Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, \({2}^{4}\) means to multiply 2 by itself 4 times, so \({2}^{4}\) means \(2\cdot 2\cdot 2\cdot 2\).

Let’s review the vocabulary for expressions with exponents.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Example

Try it.

Simplify: ⓐ \({4}^{3}\) ⓑ \({7}^{1}\) ⓒ \({(\frac{5}{6})}^{2}\) ⓓ \({(0.63)}^{2}.\)

Solution

ⓐ	\({4}^{3}\)
Multiply three factors of 4.	\(4\cdot 4\cdot 4\)
Simplify.	\(64\)
ⓑ	\({7}^{1}\)
Multiply one factor of 7.	\(7\)
ⓒ	\({(\frac{5}{6})}^{2}\)
Multiply two factors.	\((\frac{5}{6})(\frac{5}{6})\)
Simplify.	\(\frac{25}{36}\)
ⓓ	\({(0.63)}^{2}\)
Multiply two factors.	\((0.63)(0.63)\)
Simplify.	\(0.3969\)

Example

Try it.

Simplify: ⓐ \({(-5)}^{4}\) ⓑ \(\text{-}{5}^{4}.\)

Solution

ⓐ	\({(-5)}^{4}\)
Multiply four factors of \(-5\).	\((-5)(-5)(-5)(-5)\)
Simplify.	\(625\)
ⓑ	\(\text{-}{5}^{4}\)
Multiply four factors of 5.	\(\text{-}(5\cdot 5\cdot 5\cdot 5)\)
Simplify.	\(-625\)

Notice the similarities and differences in ⓐ and ⓑ ! Why are the answers different? As we follow the order of operations in part ⓐ the parentheses tell us to raise the \((-5)\) to the 4^th power. In part ⓑ we raise just the 5 to the 4^th power and then take the opposite.

Simplify Expressions Using the Product Property for Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

We’ll derive the properties of exponents by looking for patterns in several examples.

First, we will look at an example that leads to the Product Property.

What does this mean? How many factors altogether?
So, we have
Notice that 5 is the sum of the exponents, 2 and 3.

We write:

\[\begin{array}{l}{x}^{2}\cdot {x}^{3} \\ {x}^{2+3} \\ {x}^{5}\end{array}\]

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

An example with numbers helps to verify this property.

\[\begin{array}{lll}{2}^{2}\cdot {2}^{3} & \overset{?}{=} & {2}^{2+3} \\ 4\cdot 8 & \overset{?}{=} & {2}^{5} \\ 32 & = & 32\ \text{✓}\end{array}\]

Example

Try it.

Simplify: \({y}^{5}\cdot {y}^{6}.\)

Solution

Use the product property, am · an = am+n.
Simplify.

Example

Try it.

Simplify: ⓐ \({2}^{5}\cdot {2}^{9}\) ⓑ \(3\cdot {3}^{4}.\)

Solution

1. ⓐ
   

   Use the product property, am · an = am+n.
   Simplify.

2. ⓑ
   

   Use the product property, am · an = am+n.
   Simplify.

Example

Try it.

Simplify: ⓐ \({a}^{7}\cdot a\) ⓑ \({x}^{27}\cdot {x}^{13}.\)

Solution

1. ⓐ
   

   Rewrite, a = a1.
   Use the product property, am · an = am+n.
   Simplify.

2. ⓑ
   

   Notice, the bases are the same, so add the exponents.
   Simplify.

Example

Try it.

Simplify: \({d}^{4}\cdot {d}^{5}\cdot {d}^{2}.\)

Solution

Add the exponents, since bases are the same.
Simplify.

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

What does this mean? How many factors altogether?
So we have
Notice that 6 is the product of the exponents, 2 and 3.

We write:

\[\begin{array}{l}{({x}^{2})}^{3} \\ {x}^{2\cdot 3} \\ {x}^{6}\end{array}\]

We multiplied the exponents. This leads to the Power Property for Exponents.

An example with numbers helps to verify this property.

\[\begin{array}{lll}{({3}^{2})}^{3} & \overset{?}{=} & {3}^{2\cdot 3} \\ {(9)}^{3} & \overset{?}{=} & {3}^{6} \\ 729 & = & 729\ \text{✓}\end{array}\]

Example

Try it.

Simplify: ⓐ \({({y}^{5})}^{9}\) ⓑ \({({4}^{4})}^{7}.\)

Solution

ⓐ

Use the power property, (am)n = am·n.
Simplify.

ⓑ

Use the power property.
Simplify.

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

	\({(2x)}^{3}\)
What does this mean?	\(2x\cdot 2x\cdot 2x\)
We group the like factors together.	\(2\cdot 2\cdot 2\cdot x\cdot x\cdot x\)
How many factors of 2 and of \(x\)?	\({2}^{3}\cdot {x}^{3}\)

Notice that each factor was raised to the power and \({(2x)}^{3}\) is \({2}^{3}\cdot {x}^{3}\).

We write:	\({(2x)}^{3}\)
	\({2}^{3}\cdot {x}^{3}\)

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

An example with numbers helps to verify this property:

\[\begin{array}{lll}{(2\cdot 3)}^{2} & \overset{?}{=} & {2}^{2}\cdot {3}^{2} \\ {6}^{2} & \overset{?}{=} & 4\cdot 9 \\ 36 & = & 36\ \text{✓}\end{array}\]

Example

Try it.

Simplify: ⓐ \({(-9d)}^{2}\) ⓑ \({(3mn)}^{3}.\)

Solution

1. ⓐ
   

   Use Power of a Product Property, (ab)m = ambm.
   Simplify.

2. ⓑ
   

   Use Power of a Product Property, (ab)m = ambm.
   Simplify.

Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

All exponent properties hold true for any real numbers \(m\ \text{and}\ n\). Right now, we only use whole number exponents.

Example

Try it.

Simplify: ⓐ \({({y}^{3})}^{6}{({y}^{5})}^{4}\) ⓑ \({(-6{x}^{4}{y}^{5})}^{2}.\)

Solution

ⓐ	\({({y}^{3})}^{6}{({y}^{5})}^{4}\)
Use the Power Property.	\({y}^{18}\cdot {y}^{20}\)
Add the exponents.	\({y}^{38}\)
ⓑ	\({(-6{x}^{4}{y}^{5})}^{2}\)
Use the Product to a Power Property.	\({(-6)}^{2}{({x}^{4})}^{2}{({y}^{5})}^{2}\)
Use the Power Property.	\({(-6)}^{2}\)
Simplify.	\(36{x}^{8}{y}^{10}\)

Example

Try it.

Simplify: ⓐ \({(5m)}^{2}(3{m}^{3})\) ⓑ \({(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}.\)

Solution

ⓐ	\({(5m)}^{2}(3{m}^{3})\)
Raise \(5m\) to the second power.	\({5}^{2}{m}^{2}\cdot 3{m}^{3}\)
Simplify.	\(25{m}^{2}\cdot 3{m}^{3}\)
Use the Commutative Property.	\(25\cdot 3\cdot {m}^{2}\cdot {m}^{3}\)
Multiply the constants and add the exponents.	\(75{m}^{5}\)
ⓑ	\({(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}\)
Use the Product to a Power Property.	\(({3}^{4}{x}^{8}{y}^{4})({2}^{3}{x}^{3}{y}^{6})\)
Simplify.	\((81{x}^{8}{y}^{4})(8{x}^{3}{y}^{6})\)
Use the Commutative Property.	\(81\cdot 8\cdot {x}^{8}\cdot {x}^{3}\cdot {y}^{4}\cdot {y}^{6}\)
Multiply the constants and add the exponents.	\(648{x}^{11}{y}^{10}\)

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

Example

Try it.

Multiply: \((3{x}^{2})(-4{x}^{3}).\)

Solution

	\((3{x}^{2})(-4{x}^{3})\)
Use the Commutative Property to rearrange the terms.	\(3\cdot (-4)\cdot {x}^{2}\cdot {x}^{3}\)
Multiply.	\(-12{x}^{5}\)

Example

Try it.

Multiply: \((\frac{5}{6}{x}^{3}y)(12x{y}^{2}).\)

Solution

	\((\frac{5}{6}{x}^{3}y)(12x{y}^{2})\)
Use the Commutative Property to rearrange the terms.	\(\frac{5}{6}\cdot 12\cdot {x}^{3}\cdot x\cdot y\cdot {y}^{2}\)
Multiply.	\(10{x}^{4}{y}^{3}\)

Key Concepts

• Exponential Notation
• Properties of Exponents
  • If \(a,b\) are real numbers and \(m,n\) are whole numbers, then
    \(\begin{array}{llllll}\text{Product Property} & & & {a}^{m}\cdot {a}^{n} & = & {a}^{m+n} \\ \text{Power Property} & & & {({a}^{m})}^{n} & = & {a}^{m\cdot n} \\ \text{Product to a Power} & & & {(ab)}^{m} & = & {a}^{m}{b}^{m}\end{array}\)

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

Try it.

ⓐ \({3}^{5}\) ⓑ \({9}^{1}\) ⓒ \({(\frac{1}{3})}^{2}\) ⓓ \({(0.2)}^{4}\)

Try it.

ⓐ \({10}^{4}\) ⓑ \({17}^{1}\) ⓒ \({(\frac{2}{9})}^{2}\) ⓓ \({(0.5)}^{3}\)

Solution

ⓐ 10,000 ⓑ 17 ⓒ \(\frac{4}{81}\) ⓓ 0.125

Try it.

ⓐ \({2}^{6}\) ⓑ \({14}^{1}\) ⓒ \({(\frac{2}{5})}^{3}\) ⓓ \({(0.7)}^{2}\)

Try it.

ⓐ \({8}^{3}\) ⓑ \({8}^{1}\) ⓒ \({(\frac{3}{4})}^{3}\) ⓓ \({(0.4)}^{3}\)

Solution

ⓐ 512 ⓑ 8 ⓒ \(\frac{27}{64}\)
ⓓ 0.064

Try it.

ⓐ \({(-6)}^{4}\) ⓑ \(\text{-}{6}^{4}\)

Try it.

ⓐ \({(-2)}^{6}\) ⓑ \(\text{-}{2}^{6}\)

Solution

ⓐ 64 ⓑ \(-64\)

Try it.

ⓐ \(\text{-}{(\frac{1}{4})}^{4}\) ⓑ \({(-\frac{1}{4})}^{4}\)

Try it.

ⓐ \(\text{-}{(\frac{2}{3})}^{2}\) ⓑ \({(-\frac{2}{3})}^{2}\)

Solution

ⓐ \(-\frac{4}{9}\) ⓑ \(\frac{4}{9}\)

Try it.

ⓐ \(\text{-}{0.5}^{2}\) ⓑ \({(-0.5)}^{2}\)

Try it.

ⓐ \(\text{-}{0.1}^{4}\) ⓑ \({(-0.1)}^{4}\)

Solution

ⓐ \(-0.0001\) ⓑ 0.0001

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

Try it.

\({d}^{3}\cdot {d}^{6}\)

Try it.

\({x}^{4}\cdot {x}^{2}\)

Solution

\({x}^{6}\)

Try it.

\({n}^{19}\cdot {n}^{12}\)

Try it.

\({q}^{27}\cdot {q}^{15}\)

Solution

\({q}^{42}\)

Try it.

ⓐ \({4}^{5}\cdot {4}^{9}\) ⓑ \({8}^{9}\cdot 8\)

Try it.

ⓐ \({3}^{10}\cdot {3}^{6}\) ⓑ \(5\cdot {5}^{4}\)

Solution

ⓐ \({3}^{16}\) ⓑ \({5}^{5}\)

Try it.

ⓐ \(y\cdot {y}^{3}\) ⓑ \({z}^{25}\cdot {z}^{8}\)

Try it.

ⓐ \({w}^{5}\cdot {w}^{}\) ⓑ \({u}^{41}\cdot {u}^{53}\)

Solution

ⓐ \({w}^{6}\) ⓑ \({u}^{94}\)

Try it.

\(w\cdot {w}^{2}\cdot {w}^{3}\)

Try it.

\(y\cdot {y}^{3}\cdot {y}^{5}\)

Solution

\({y}^{9}\)

Try it.

\({a}^{4}\cdot {a}^{3}\cdot {a}^{9}\)

Try it.

\({c}^{5}\cdot {c}^{11}\cdot {c}^{2}\)

Solution

\({c}^{18}\)

Try it.

\({m}^{x}\cdot {m}^{3}\)

Try it.

\({n}^{y}\cdot {n}^{2}\)

Solution

\({n}^{y+2}\)

Try it.

\({y}^{a}\cdot {y}^{b}\)

Try it.

\({x}^{p}\cdot {x}^{q}\)

Solution

\({x}^{p+q}\)

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression using the Power Property for Exponents.

Try it.

ⓐ \({({m}^{4})}^{2}\) ⓑ \({({10}^{3})}^{6}\)

Try it.

ⓐ \({({b}^{2})}^{7}\) ⓑ \({({3}^{8})}^{2}\)

Solution

ⓐ \({b}^{14}\) ⓑ \({3}^{16}\)

Try it.

ⓐ \({({y}^{3})}^{x}\) ⓑ \({({5}^{x})}^{y}\)

Try it.

ⓐ \({({x}^{2})}^{y}\) ⓑ \({({7}^{a})}^{b}\)

Solution

ⓐ \({x}^{2y}\) ⓑ \({7}^{ab}\)

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

Try it.

ⓐ \({(6a)}^{2}\) ⓑ \({(3xy)}^{2}\)

Try it.

ⓐ \({(5x)}^{2}\) ⓑ \({(4ab)}^{2}\)

Solution

ⓐ \(25{x}^{2}\) ⓑ \(16{a}^{2}{b}^{2}\)

Try it.

ⓐ \({(-4m)}^{3}\) ⓑ \({(5ab)}^{3}\)

Try it.

ⓐ \({(-7n)}^{3}\) ⓑ \({(3xyz)}^{4}\)

Solution

ⓐ \(-343{n}^{3}\) ⓑ \(81{x}^{4}{y}^{4}{z}^{4}\)

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Try it.

ⓐ \({({y}^{2})}^{4}\cdot {({y}^{3})}^{2}\) ⓑ \({(10{a}^{2}b)}^{3}\)

Try it.

ⓐ \({({w}^{4})}^{3}\cdot {({w}^{5})}^{2}\) ⓑ \({(2x{y}^{4})}^{5}\)

Solution

ⓐ \({w}^{22}\) ⓑ \(32{x}^{5}{y}^{20}\)

Try it.

ⓐ \({(-2{r}^{3}{s}^{2})}^{4}\) ⓑ \({({m}^{5})}^{3}\cdot {({m}^{9})}^{4}\)

Try it.

ⓐ \({(-10{q}^{2}{p}^{4})}^{3}\) ⓑ \({({n}^{3})}^{10}\cdot {({n}^{5})}^{2}\)

Solution

ⓐ \(-1000{q}^{6}{p}^{12}\) ⓑ \({n}^{40}\)

Try it.

ⓐ \({(3x)}^{2}(5x)\) ⓑ \({(5{t}^{2})}^{3}{(3t)}^{2}\)

Try it.

ⓐ \({(2y)}^{3}(6y)\) ⓑ \({(10{k}^{4})}^{3}{(5{k}^{6})}^{2}\)

Solution

ⓐ \(48{y}^{4}\) ⓑ \(25,000{k}^{24}\)

Try it.

ⓐ \({(5a)}^{2}{(2a)}^{3}\) ⓑ \({(\frac{1}{2}{y}^{2})}^{3}{(\frac{2}{3}y)}^{2}\)

Try it.

ⓐ \({(4b)}^{2}{(3b)}^{3}\) ⓑ \({(\frac{1}{2}{j}^{2})}^{5}{(\frac{2}{5}{j}^{3})}^{2}\)

Solution

ⓐ \(432{b}^{5}\) ⓑ \(\frac{1}{200}{j}^{16}\)

Try it.

ⓐ \({(\frac{2}{5}{x}^{2}y)}^{3}\) ⓑ \({(\frac{8}{9}x{y}^{4})}^{2}\)

Try it.

ⓐ \({(2{r}^{2})}^{3}{(4r)}^{2}\) ⓑ \({(3{x}^{3})}^{3}{({x}^{5})}^{4}\)

Solution

ⓐ \(128{r}^{8}\) ⓑ \({27x}^{29}\)

Try it.

ⓐ \({({m}^{2}n)}^{2}{(2m{n}^{5})}^{4}\) ⓑ \({(3p{q}^{4})}^{2}{(6{p}^{6}q)}^{2}\)

Multiply Monomials

In the following exercises, multiply the monomials.

Try it.

\((6{y}^{7})(-3{y}^{4})\)

Solution

\(-18{y}^{11}\)

Try it.

\((-10{x}^{5})(-3{x}^{3})\)

Try it.

\((-8{u}^{6})(-9u)\)

Solution

\(72{u}^{7}\)

Try it.

\((-6{c}^{4})(-12c)\)

Try it.

\((\frac{1}{5}{f}^{8})(20{f}^{3})\)

Solution

\(4{f}^{11}\)

Try it.

\((\frac{1}{4}{d}^{5})(36{d}^{2})\)

Try it.

\((4{a}^{3}b)(9{a}^{2}{b}^{6})\)

Solution

\(36{a}^{5}{b}^{7}\)

Try it.

\((6{m}^{4}{n}^{3})(7m{n}^{5})\)

Try it.

\((\frac{4}{7}r{s}^{2})(14r{s}^{3})\)

Solution

\(8{r}^{2}{s}^{5}\)

Try it.

\((\frac{5}{8}{x}^{3}y)(24{x}^{5}y)\)

Try it.

\((\frac{2}{3}{x}^{2}y)(\frac{3}{4}x{y}^{2})\)

Solution

\(\frac{1}{2}{x}^{3}{y}^{3}\)

Try it.

\((\frac{3}{5}{m}^{3}{n}^{2})(\frac{5}{9}{m}^{2}{n}^{3})\)

Mixed Practice

In the following exercises, simplify each expression.

Try it.

\({({x}^{2})}^{4}\cdot {({x}^{3})}^{2}\)

Solution

\({x}^{14}\)

Try it.

\({({y}^{4})}^{3}\cdot {({y}^{5})}^{2}\)

Try it.

\({({a}^{2})}^{6}\cdot {({a}^{3})}^{8}\)

Solution

\({a}^{36}\)

Try it.

\({({b}^{7})}^{5}\cdot {({b}^{2})}^{6}\)

Try it.

\({(2{m}^{6})}^{3}\)

Solution

\(8{m}^{18}\)

Try it.

\({(3{y}^{2})}^{4}\)

Try it.

\({(10{x}^{2}y)}^{3}\)

Solution

\(1000{x}^{6}{y}^{3}\)

Try it.

\({(2m{n}^{4})}^{5}\)

Try it.

\({(-2{a}^{3}{b}^{2})}^{4}\)

Solution

\(16{a}^{12}{b}^{8}\)

Try it.

\({(-10{u}^{2}{v}^{4})}^{3}\)

Try it.

\({(\frac{2}{3}{x}^{2}y)}^{3}\)

Solution

\(\frac{8}{27}{x}^{6}{y}^{3}\)

Try it.

\({(\frac{7}{9}p{q}^{4})}^{2}\)

Try it.

\({(8{a}^{3})}^{2}{(2a)}^{4}\)

Solution

\(1024{a}^{10}\)

Try it.

\({(5{r}^{2})}^{3}{(3r)}^{2}\)

Try it.

\({(10{p}^{4})}^{3}{(5{p}^{6})}^{2}\)

Solution

\(25000{p}^{24}\)

Try it.

\({(4{x}^{3})}^{3}{(2{x}^{5})}^{4}\)

Try it.

\({(\frac{1}{2}{x}^{2}{y}^{3})}^{4}{(4{x}^{5}{y}^{3})}^{2}\)

Solution

\({x}^{18}{y}^{18}\)

Try it.

\({(\frac{1}{3}{m}^{3}{n}^{2})}^{4}{(9{m}^{8}{n}^{3})}^{2}\)

Try it.

\({(3{m}^{2}n)}^{2}{(2m{n}^{5})}^{4}\)

Solution

\(144{m}^{8}{n}^{22}\)

Try it.

\({(2p{q}^{4})}^{3}{(5{p}^{6}q)}^{2}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, \({2}^{4}\) means to multiply four factors of \(2,\) so \({2}^{4}\) means \(2\cdot 2\cdot 2\cdot 2.\) This format is known as exponential notation.

In the expression \({a}^{m},\) the exponent tells us how many times we use the base \(a\) as a factor.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Example

Try it.

Simplify:

1. ⓐ \(\ {5}^{3}\)
2. ⓑ \(\ {9}^{1}\)

Solution

ⓐ
	\({5}^{3}\)
Multiply 3 factors of 5.	\(5\cdot 5\cdot 5\)
Simplify.	\(125\)

ⓑ
	\({9}^{1}\)
Multiply 1 factor of 9.	\(9\)

Example

Try it.

Simplify:

1. ⓐ \(\ {(\frac{7}{8})}^{2}\)
2. ⓑ \(\ {(0.74)}^{2}\)

Solution

ⓐ
	\({(\frac{7}{8})}^{2}\)
Multiply two factors.	\((\frac{7}{8})(\frac{7}{8})\)
Simplify.	\(\frac{49}{64}\)

ⓑ
	\({(0.74)}^{2}\)
Multiply two factors.	\((0.74)(0.74)\)
Simplify.	\(0.5476\)

Example

Try it.

Simplify:

1. ⓐ \(\ {(-3)}^{4}\)
2. ⓑ \(\ {-3}^{4}\)

Solution

ⓐ
	\({(-3)}^{4}\)
Multiply four factors of −3.	\((-3)(-3)(-3)(-3)\)
Simplify.	\(81\)

ⓑ
	\({-3}^{4}\)
Multiply two factors.	\(-(3\cdot 3\cdot 3\cdot 3)\)
Simplify.	\(-81\)

Notice the similarities and differences in parts ⓐ and ⓑ. Why are the answers different? In part ⓐ the parentheses tell us to raise the (−3) to the 4^th power. In part ⓑ we raise only the 3 to the 4^th power and then find the opposite.

Simplify Expressions Using the Product Property of Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

First, we will look at an example that leads to the Product Property.

What does this mean? How many factors altogether?
So, we have
Notice that 5 is the sum of the exponents, 2 and 3.
We write:	\({x}^{2}⋅{x}^{3}\) \({x}^{2+3}\) \({x}^{5}\)

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

An example with numbers helps to verify this property.

\[\begin{array}{lll}{2}^{2}\cdot {2}^{3} & \overset{?}{=} & {2}^{2+3} \\ 4\cdot 8 & \overset{?}{=} & {2}^{5} \\ 32 & = & 32✓\end{array}\]

Example

Try it.

Simplify: \({x}^{5}\cdot {x}^{7}.\)

Solution

	\({x}^{5}\cdot {x}^{7}\)
Use the product property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify.	\({x}^{12}\)

Example

Try it.

Simplify: \({b}^{4}\cdot b.\)

Solution

	\({b}^{4}\cdot b\)
Rewrite, \(b={b}^{1}.\)	\({b}^{4}\cdot {b}^{1}\)
Use the product property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify.	\({b}^{5}\)

Example

Try it.

Simplify: \({2}^{7}\cdot {2}^{9}.\)

Solution

	\({2}^{7}\cdot {2}^{9}\)
Use the product property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify.	\({2}^{16}\)

Example

Try it.

Simplify: \({y}^{17}\cdot {y}^{23}.\)

Solution

	\({y}^{17}\cdot {y}^{23}\)
Notice, the bases are the same, so add the exponents.
Simplify.	\({y}^{40}\)

We can extend the Product Property of Exponents to more than two factors.

Example

Try it.

Simplify: \({x}^{3}\cdot {x}^{4}\cdot {x}^{2}.\)

Solution

	\({x}^{3}\cdot {x}^{4}\cdot {x}^{2}\)
Add the exponents, since the bases are the same.
Simplify.	\({x}^{9}\)

Simplify Expressions Using the Power Property of Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

What does this mean? How many factors altogether?
So, we have
Notice that 6 is the product of the exponents, 2 and 3.
We write:	\({({x}^{2})}^{3}\) \({x}^{2⋅3}\) \({x}^{6}\)

We multiplied the exponents. This leads to the Power Property for Exponents.

An example with numbers helps to verify this property.

\[\begin{array}{lll}{({5}^{2})}^{3} & \overset{?}{=} & {5}^{2\cdot 3} \\ {(25)}^{3} & \overset{?}{=} & {5}^{6} \\ 15,625 & = & 15,625✓\end{array}\]

Example

Try it.

Simplify:

1. ⓐ \(\ {({x}^{5})}^{7}\)
2. ⓐ \(\ {({3}^{6})}^{8}\)

Solution

ⓐ
	\({({x}^{5})}^{7}\)
Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\)
Simplify.	\({x}^{35}\)

ⓑ
	\({({3}^{6})}^{8}\)
Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\)
Simplify.	\({3}^{48}\)

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

	\({(2x)}^{3}\)
What does this mean?	\(2x\cdot 2x\cdot 2x\)
We group the like factors together.	\(2\cdot 2\cdot 2\cdot x\cdot x\cdot x\)
How many factors of 2 and of \(x?\)	\({2}^{3}\cdot {x}^{3}\)
Notice that each factor was raised to the power.	\({(2x)}^{3}\ \text{is}\ {2}^{3}\cdot {x}^{3}\)
We write:	\({(2x)}^{3}\) \({2}^{3}\cdot {x}^{3}\)

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

An example with numbers helps to verify this property:

\[\begin{array}{lll}{(2\cdot 3)}^{2} & \overset{?}{=} & {2}^{2}\cdot {3}^{2} \\ {6}^{2} & \overset{?}{=} & 4\cdot 9 \\ 36 & = & 36✓\end{array}\]

Example

Try it.

Simplify: \({(-11x)}^{2}.\)

Solution

	\({(-11x)}^{2}\)
Use the Power of a Product Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)
Simplify.	\(121{x}^{2}\)

Example

Try it.

Simplify: \({(3xy)}^{3}.\)

Solution

	\({(3xy)}^{3}\)
Raise each factor to the third power.
Simplify.	\(27{x}^{3}{y}^{3}\)

Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Example

Try it.

Simplify: \({({x}^{2})}^{6}{({x}^{5})}^{4}.\)

Solution

	\({({x}^{2})}^{6}{({x}^{5})}^{4}\)
Use the Power Property.	\({x}^{12}\cdot {x}^{20}\)
Add the exponents.	\({x}^{32}\)

Example

Try it.

Simplify: \({(-7{x}^{3}{y}^{4})}^{2}.\)

Solution

	\({(-7{x}^{3}{y}^{4})}^{2}\)
Take each factor to the second power.	\({(-7)}^{2}{({x}^{3})}^{2}{({y}^{4})}^{2}\)
Use the Power Property.	\(49{x}^{6}{y}^{8}\)

Example

Try it.

Simplify: \({(6n)}^{2}(4{n}^{3}).\)

Solution

	\({(6n)}^{2}(4{n}^{3})\)
Raise \(6n\) to the second power.	\({6}^{2}{n}^{2}\cdot 4{n}^{3}\)
Simplify.	\(36{n}^{2}\cdot 4{n}^{3}\)
Use the Commutative Property.	\(36\cdot 4\cdot {n}^{2}\cdot {n}^{3}\)
Multiply the constants and add the exponents.	\(144{n}^{5}\)

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.

Example

Try it.

Simplify: \({(3{p}^{2}q)}^{4}{(2p{q}^{2})}^{3}.\)

Solution

	\({(3{p}^{2}q)}^{4}{(2p{q}^{2})}^{3}\)
Use the Power of a Product Property.	\({3}^{4}{({p}^{2})}^{4}{q}^{4}\cdot {2}^{3}{p}^{3}{({q}^{2})}^{3}\)
Use the Power Property.	\(81{p}^{8}{q}^{4}\cdot 8{p}^{3}{q}^{6}\)
Use the Commutative Property.	\(81\cdot 8\cdot {p}^{8}\cdot {p}^{3}\cdot {q}^{4}\cdot {q}^{6}\)
Multiply the constants and add the exponents for each variable.	\(648{p}^{11}{q}^{10}\)

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Example

Try it.

Multiply: \((4{x}^{2})(-5{x}^{3}).\)

Solution

	\((4{x}^{2})(-5{x}^{3})\)
Use the Commutative Property to rearrange the factors.	\(4\cdot (-5)\cdot {x}^{2}\cdot {x}^{3}\)
Multiply.	\(-20{x}^{5}\)

Example

Try it.

Multiply: \((\frac{3}{4}\ {c}^{3}d)(12c{d}^{2}).\)

Solution

	\((\frac{3}{4}\ {c}^{3}d)(12c{d}^{2})\)
Use the Commutative Property to rearrange the factors.	\(\frac{3}{4}\cdot 12\cdot {c}^{3}\cdot c\cdot d\cdot {d}^{2}\)
Multiply.	\(9{c}^{4}{d}^{3}\)

Key Concepts

• Exponential Notation

  This is read \(a\) to the \({m}^{\text{th}}\) power.

• Product Property of Exponents
  • If \(a\) is a real number and \(m,n\) are counting numbers, then \[{a}^{m}\cdot {a}^{n}={a}^{m+n}\]
  • To multiply with like bases, add the exponents.
• Power Property for Exponents
  • If \(a\) is a real number and \(m,n\) are counting numbers, then \[{({a}^{m})}^{n}={a}^{m⋅n}\]
• Product to a Power Property for Exponents
  • If \(a\) and \(b\) are real numbers and \(m\) is a whole number, then\[{(ab)}^{m}={a}^{m}{b}^{m}\]

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

Try it.

\({10}^{3}\)

Try it.

\({(\frac{1}{2})}^{2}\)

Solution

\(\frac{1}{4}\)

Try it.

\({(\frac{3}{5})}^{2}\)

Try it.

\({(0.2)}^{3}\)

Solution

0.008

Try it.

\({(0.4)}^{3}\)

Try it.

\({(-5)}^{4}\)

Solution

625

Try it.

\({(-3)}^{5}\)

Try it.

\({-5}^{4}\)

Solution

−625

Try it.

\({-3}^{5}\)

Try it.

\({-10}^{4}\)

Solution

−10,000

Try it.

\({-2}^{6}\)

Try it.

\({(-\frac{2}{3})}^{3}\)

Solution

\(-\frac{8}{27}\)

Try it.

\({(-\frac{1}{4})}^{4}\)

Try it.

\(-{0.5}^{2}\)

Solution

−0.25

Try it.

\(-{0.1}^{4}\)

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression using the Product Property of Exponents.

Try it.

\({x}^{3}\cdot {x}^{6}\)

Solution

x⁹

Try it.

\({m}^{4}\cdot {m}^{2}\)

Try it.

\(a\cdot {a}^{4}\)

Solution

a⁵

Try it.

\({y}^{12}\cdot y\)

Try it.

\({3}^{5}\cdot {3}^{9}\)

Solution

3¹⁴

Try it.

\({5}^{10}\cdot {5}^{6}\)

Try it.

\(z\cdot {z}^{2}\cdot {z}^{3}\)

Solution

z⁶

Try it.

\(a\cdot {a}^{3}\cdot {a}^{5}\)

Try it.

\({x}^{a}\cdot {x}^{2}\)

Solution

x^a+2

Try it.

\({y}^{p}\cdot {y}^{3}\)

Try it.

\({y}^{a}\cdot {y}^{b}\)

Solution

y^a+b

Try it.

\({x}^{p}\cdot {x}^{q}\)

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression using the Power Property of Exponents.

Try it.

\({({u}^{4})}^{2}\)

Solution

u⁸

Try it.

\({({x}^{2})}^{7}\)

Try it.

\({({y}^{5})}^{4}\)

Solution

y²⁰

Try it.

\({({a}^{3})}^{2}\)

Try it.

\({({10}^{2})}^{6}\)

Solution

10¹²

Try it.

\({({2}^{8})}^{3}\)

Try it.

\({({x}^{15})}^{6}\)

Solution

x⁹⁰

Try it.

\({({y}^{12})}^{8}\)

Try it.

\({({x}^{2})}^{y}\)

Solution

x^2y

Try it.

\({({y}^{3})}^{x}\)

Try it.

\({({5}^{x})}^{y}\)

Solution

5^xy

Try it.

\({({7}^{a})}^{b}\)

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

Try it.

\({(5a)}^{2}\)

Solution

25a²

Try it.

\({(7x)}^{2}\)

Try it.

\({(-6m)}^{3}\)

Solution

−216m³

Try it.

\({(-9n)}^{3}\)

Try it.

\({(4rs)}^{2}\)

Solution

16r²s²

Try it.

\({(5ab)}^{3}\)

Try it.

\({(4xyz)}^{4}\)

Solution

256x⁴y⁴z⁴

Try it.

\({(-5abc)}^{3}\)

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Try it.

\({({x}^{2})}^{4}\cdot {({x}^{3})}^{2}\)

Solution

x¹⁴

Try it.

\({({y}^{4})}^{3}\cdot {({y}^{5})}^{2}\)

Try it.

\({({a}^{2})}^{6}\cdot {({a}^{3})}^{8}\)

Solution

a³⁶

Try it.

\({({b}^{7})}^{5}\cdot {({b}^{2})}^{6}\)

Try it.

\({(3x)}^{2}(5x)\)

Solution

45x³

Try it.

\({(2y)}^{3}(6y)\)

Try it.

\({(5a)}^{2}{(2a)}^{3}\)

Solution

200a⁵

Try it.

\({(4b)}^{2}{(3b)}^{3}\)

Try it.

\({(2{m}^{6})}^{3}\)

Solution

8m¹⁸

Try it.

\({(3{y}^{2})}^{4}\)

Try it.

\({(10{x}^{2}y)}^{3}\)

Solution

1,000x⁶y³

Try it.

\({(2m{n}^{4})}^{5}\)

Try it.

\({(-2{a}^{3}{b}^{2})}^{4}\)

Solution

16a¹²b⁸

Try it.

\({(-10{u}^{2}{v}^{4})}^{3}\)

Try it.

\({(\frac{2}{3}\ {x}^{2}y)}^{3}\)

Solution

\(\frac{8}{27}\ {x}^{6}{y}^{3}\)

Try it.

\({(\frac{7}{9}\ p{q}^{4})}^{2}\)

Try it.

\({(8{a}^{3})}^{2}{(2a)}^{4}\)

Solution

1,024a¹⁰

Try it.

\({(5{r}^{2})}^{3}{(3r)}^{2}\)

Try it.

\({(10{p}^{4})}^{3}{(5{p}^{6})}^{2}\)

Solution

25,000p²⁴

Try it.

\({(4{x}^{3})}^{3}{(2{x}^{5})}^{4}\)

Try it.

\({(\frac{1}{2}\ {x}^{2}{y}^{3})}^{4}{(4{x}^{5}{y}^{3})}^{2}\)

Solution

x¹⁸y¹⁸

Try it.

\({(\frac{1}{3}\ {m}^{3}{n}^{2})}^{4}{(9{m}^{8}{n}^{3})}^{2}\)

Try it.

\({(3{m}^{2}n)}^{2}{(2m{n}^{5})}^{4}\)

Solution

144m⁸n²²

Try it.

\({(2p{q}^{4})}^{3}{(5{p}^{6}q)}^{2}\)

Multiply Monomials

In the following exercises, multiply the following monomials.

Try it.

\((12{x}^{2})(-5{x}^{4})\)

Solution

−60x⁶

Try it.

\((-10{y}^{3})(7{y}^{2})\)

Try it.

\((-8{u}^{6})(-9u)\)

Solution

72u⁷

Try it.

\((-6{c}^{4})(-12c)\)

Try it.

\((\frac{1}{5}\ {r}^{8})(20{r}^{3})\)

Solution

4r¹¹

Try it.

\((\frac{1}{4}\ {a}^{5})(36{a}^{2})\)

Try it.

\((4{a}^{3}b)(9{a}^{2}{b}^{6})\)

Solution

36a⁵b⁷

Try it.

\((6{m}^{4}{n}^{3})(7m{n}^{5})\)

Try it.

\((\frac{4}{7}\ x{y}^{2})(14x{y}^{3})\)

Solution

8x²y⁵

Try it.

\((\frac{5}{8}\ {u}^{3}v)(24{u}^{5}v)\)

Try it.

\((\frac{2}{3}\ {x}^{2}y)(\frac{3}{4}\ x{y}^{2})\)

Solution

\(\frac{1}{2}\ {x}^{3}{y}^{3}\)

Try it.

\((\frac{3}{5}\ {m}^{3}{n}^{2})(\frac{5}{9}\ {m}^{2}{n}^{3})\)

Condensed: the full section is in OpenStax Prealgebra 2e.

اب آپ هيک کليچر اس کو حل نہیں کر سکتا ، ليکن اس کے ٹکڑے حسابي هے. نيچے سے کو ئي کوشش کريں يا خود کو ٹائپ کريں

⌨ قسم ✎ اس کو بناؤ

آپ کا ریاضی کا مسئلہ

π حل

تربيت (40)

هر کيس پر اول آزمائيں جواب کو چيک کر نے کے ليے ظاہر کريں جو صحيح هے وہ حل کر نے کے ليے کل قدم کے ليے کھولي جا سکتے هيں

1. Simplify: \(\frac{3}{4}\cdot \frac{3}{4}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   \(\frac{3}{4}\cdot \frac{3}{4}\)

2. Simplify: \((-2)(-2)(-2).\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   \(-8\)

3. Simplify: ⓐ \({4}^{3}\) ⓑ \({7}^{1}\) ⓒ \({(\frac{5}{6})}^{2}\) ⓓ \({(0.63)}^{2}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   ⓐ	\({4}^{3}\)
   Multiply three factors of 4.	\(4\cdot 4\cdot 4\)
   Simplify.	\(64\)
   ⓑ	\({7}^{1}\)
   Multiply one factor of 7.	\(7\)
   ⓒ	\({(\frac{5}{6})}^{2}\)
   Multiply two factors.	\((\frac{5}{6})(\frac{5}{6})\)
   Simplify.	\(\frac{25}{36}\)
   ⓓ	\({(0.63)}^{2}\)
   Multiply two factors.	\((0.63)(0.63)\)
   Simplify.	\(0.3969\)

4. Simplify: ⓐ \({6}^{3}\) ⓑ \({15}^{1}\) ⓒ \({(\frac{3}{7})}^{2}\) ⓓ \({(0.43)}^{2}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   ⓐ 216 ⓑ \(15\) ⓒ \(\frac{9}{49}\) ⓓ 0.1849

5. Simplify: ⓐ \({2}^{5}\) ⓑ \({21}^{1}\) ⓒ \({(\frac{2}{5})}^{3}\) ⓓ \({(0.218)}^{2}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   ⓐ \(32\) ⓑ 21 ⓒ \(\frac{8}{125}\) ⓓ \(0.047524\)

6. Simplify: ⓐ \({(-5)}^{4}\) ⓑ \(\text{-}{5}^{4}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   ⓐ	\({(-5)}^{4}\)
   Multiply four factors of \(-5\).	\((-5)(-5)(-5)(-5)\)
   Simplify.	\(625\)
   ⓑ	\(\text{-}{5}^{4}\)
   Multiply four factors of 5.	\(\text{-}(5\cdot 5\cdot 5\cdot 5)\)
   Simplify.	\(-625\)

7. Simplify: ⓐ \({(-3)}^{4}\) ⓑ \(\text{-}{3}^{4}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   ⓐ \(81\) ⓑ \(-81\)

8. Simplify: ⓐ \({(-13)}^{2}\) ⓑ \(\text{-}{13}^{2}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   ⓐ \(169\) ⓑ \(-169\)

9. Simplify: \({y}^{5}\cdot {y}^{6}.\)

   اس کے ذریعے کام اشارہ

   جواب کھوليں

   Use the product property, am · an = am+n.
   Simplify.

10. Simplify: \({b}^{9}\cdot {b}^{8}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \({b}^{17}\)

11. Simplify: \({x}^{12}\cdot {x}^{4}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \({x}^{16}\)

12. Simplify: ⓐ \({2}^{5}\cdot {2}^{9}\) ⓑ \(3\cdot {3}^{4}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    1. ⓐ
       

       Use the product property, am · an = am+n.
       Simplify.

    2. ⓑ
       

       Use the product property, am · an = am+n.
       Simplify.

13. Simplify: ⓐ \(5\cdot {5}^{5}\) ⓑ \({4}^{9}\cdot {4}^{9}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \({5}^{6}\) ⓑ \({4}^{18}\)

14. Simplify: ⓐ \({7}^{6}\cdot {7}^{8}\) ⓑ \(10\cdot {10}^{10}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \({7}^{14}\) ⓑ \({10}^{11}\)

15. Simplify: ⓐ \({a}^{7}\cdot a\) ⓑ \({x}^{27}\cdot {x}^{13}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    1. ⓐ
       

       Rewrite, a = a1.
       Use the product property, am · an = am+n.
       Simplify.

    2. ⓑ
       

       Notice, the bases are the same, so add the exponents.
       Simplify.

16. Simplify: ⓐ \({p}^{5}\cdot p\) ⓑ \({y}^{14}\cdot {y}^{29}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \({p}^{6}\) ⓑ \({y}^{43}\)

17. Simplify: ⓐ \(z\cdot {z}^{7}\) ⓑ \({b}^{15}\cdot {b}^{34}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \({z}^{8}\) ⓑ \({b}^{49}\)

18. Simplify: \({d}^{4}\cdot {d}^{5}\cdot {d}^{2}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    Add the exponents, since bases are the same.
    Simplify.

19. Simplify: \({x}^{6}\cdot {x}^{4}\cdot {x}^{8}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \({x}^{18}\)

20. Simplify: \({b}^{5}\cdot {b}^{9}\cdot {b}^{5}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \({b}^{19}\)

21. Simplify: ⓐ \({({y}^{5})}^{9}\) ⓑ \({({4}^{4})}^{7}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ
    

    Use the power property, (am)n = am·n.
    Simplify.

    
    
    ⓑ
    

    Use the power property.
    Simplify.

22. Simplify: ⓐ \({({b}^{7})}^{5}\) ⓑ \({({5}^{4})}^{3}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \({b}^{35}\) ⓑ \({5}^{12}\)

23. Simplify: ⓐ \({({z}^{6})}^{9}\) ⓑ \({({3}^{7})}^{7}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \({z}^{54}\) ⓑ \({3}^{49}\)

24. Simplify: ⓐ \({(-9d)}^{2}\) ⓑ \({(3mn)}^{3}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    1. ⓐ
       

       Use Power of a Product Property, (ab)m = ambm.
       Simplify.

    2. ⓑ
       

       Use Power of a Product Property, (ab)m = ambm.
       Simplify.

25. Simplify: ⓐ \({(-12y)}^{2}\) ⓑ \({(2wx)}^{5}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \(144{y}^{2}\) ⓑ \(32{w}^{5}{x}^{5}\)

26. Simplify: ⓐ \({(5wx)}^{3}\) ⓑ \({(-3y)}^{3}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \(125{w}^{3}{x}^{3}\) ⓑ \(-27{y}^{3}\)

27. Simplify: ⓐ \({({y}^{3})}^{6}{({y}^{5})}^{4}\) ⓑ \({(-6{x}^{4}{y}^{5})}^{2}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ	\({({y}^{3})}^{6}{({y}^{5})}^{4}\)
    Use the Power Property.	\({y}^{18}\cdot {y}^{20}\)
    Add the exponents.	\({y}^{38}\)
    ⓑ	\({(-6{x}^{4}{y}^{5})}^{2}\)
    Use the Product to a Power Property.	\({(-6)}^{2}{({x}^{4})}^{2}{({y}^{5})}^{2}\)
    Use the Power Property.	\({(-6)}^{2}\)
    Simplify.	\(36{x}^{8}{y}^{10}\)

28. Simplify: ⓐ \({({a}^{4})}^{5}{({a}^{7})}^{4}\) ⓑ \({(-2{c}^{4}{d}^{2})}^{3}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \({a}^{48}\) ⓑ \(-8{c}^{12}{d}^{6}\)

29. Simplify: ⓐ \({(-3{x}^{6}{y}^{7})}^{4}\) ⓑ \({({q}^{4})}^{5}{({q}^{3})}^{3}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \(81{x}^{24}{y}^{28}\) ⓑ \({q}^{29}\)

30. Simplify: ⓐ \({(5m)}^{2}(3{m}^{3})\) ⓑ \({(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ	\({(5m)}^{2}(3{m}^{3})\)
    Raise \(5m\) to the second power.	\({5}^{2}{m}^{2}\cdot 3{m}^{3}\)
    Simplify.	\(25{m}^{2}\cdot 3{m}^{3}\)
    Use the Commutative Property.	\(25\cdot 3\cdot {m}^{2}\cdot {m}^{3}\)
    Multiply the constants and add the exponents.	\(75{m}^{5}\)
    ⓑ	\({(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}\)
    Use the Product to a Power Property.	\(({3}^{4}{x}^{8}{y}^{4})({2}^{3}{x}^{3}{y}^{6})\)
    Simplify.	\((81{x}^{8}{y}^{4})(8{x}^{3}{y}^{6})\)
    Use the Commutative Property.	\(81\cdot 8\cdot {x}^{8}\cdot {x}^{3}\cdot {y}^{4}\cdot {y}^{6}\)
    Multiply the constants and add the exponents.	\(648{x}^{11}{y}^{10}\)

31. Simplify: ⓐ \({(5n)}^{2}(3{n}^{10})\) ⓑ \({({c}^{4}{d}^{2})}^{5}{(3c{d}^{5})}^{4}.\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \(75{n}^{12}\) ⓑ \(81{c}^{24}{d}^{30}\)

32. Simplify: ⓐ \({({a}^{3}{b}^{2})}^{6}{(4a{b}^{3})}^{4}\) ⓑ \({(2x)}^{3}(5{x}^{7}).\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ \(256{a}^{22}{b}^{24}\) ⓑ \(40{x}^{10}\)

33. Multiply: \((3{x}^{2})(-4{x}^{3}).\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    	\((3{x}^{2})(-4{x}^{3})\)
    Use the Commutative Property to rearrange the terms.	\(3\cdot (-4)\cdot {x}^{2}\cdot {x}^{3}\)
    Multiply.	\(-12{x}^{5}\)

34. Multiply: \((5{y}^{7})(-7{y}^{4}).\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \(-35{y}^{11}\)

35. Multiply: \((-6{b}^{4})(-9{b}^{5}).\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \(54{b}^{9}\)

36. Multiply: \((\frac{5}{6}{x}^{3}y)(12x{y}^{2}).\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    	\((\frac{5}{6}{x}^{3}y)(12x{y}^{2})\)
    Use the Commutative Property to rearrange the terms.	\(\frac{5}{6}\cdot 12\cdot {x}^{3}\cdot x\cdot y\cdot {y}^{2}\)
    Multiply.	\(10{x}^{4}{y}^{3}\)

37. Multiply: \((\frac{2}{5}{a}^{4}{b}^{3})(15a{b}^{3}).\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \(6{a}^{5}{b}^{6}\)

38. Multiply: \((\frac{2}{3}{r}^{5}s)(12{r}^{6}{s}^{7}).\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    \(8{r}^{11}{s}^{8}\)

39. ⓐ \({3}^{5}\) ⓑ \({9}^{1}\) ⓒ \({(\frac{1}{3})}^{2}\) ⓓ \({(0.2)}^{4}\)

    اس کے ذریعے کام اشارہ

40. ⓐ \({10}^{4}\) ⓑ \({17}^{1}\) ⓒ \({(\frac{2}{9})}^{2}\) ⓓ \({(0.5)}^{3}\)

    اس کے ذریعے کام اشارہ

    جواب کھوليں

    ⓐ 10,000 ⓑ 17 ⓒ \(\frac{4}{81}\) ⓓ 0.125

آپ کے اپنے کام رکھیں

مفت اکاؤنٹ تمام درسوں پر نوٹ شامل کر تا هے ، آپ نے جو ختم کر يا هے اس کا ريکورڈ ، آپ کے حل کيے گئے مسئلے ایک جگہ پر اور آپ اس صفحے کے بارے ميں سوال کر سکتے هيں تو تيرچر ريتھ ميں خود تمام لوگوں کے ليے کھلا هے ، خواہ آپ داخل هے يا نہيں

رجسٹر لاگ ان

علامتیں

تمام تعريف کے ليے ، تصوير کے ليے اور اس ميں ہر حروف کا مطلب کے ليے کو ئي علامت کو ٹک کريں

\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

کیسے: Use Multiplication Properties of Exponents

1. Simplify expressions with exponents
2. Simplify expressions using the Product Property of Exponents
3. Simplify expressions using the Power Property of Exponents
4. Simplify expressions using the Product to a Power Property
5. Simplify expressions by applying several properties
6. Multiply monomials
7. If
8. To multiply with like bases, add the exponents.

سوالات لوگ پوچھتے ہیں

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

اس صفحے کے حصے سے تناسب میں لایا گیا ہے OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). یہاں سے کڑوا اور دوبارہ بیان کیا گیا ہے؛ غلطیاں ہماری ہیں. اسکے ذريعے کو غير تجاري استعمال کے ليے لائسنس ديے گيا هے اس ليکھے کو مفت ميں ملتا هے اور اسکے ليے کوئی خرچ نہیں کريے جا تا هے اور اس کو عين لائسنس کے تحت شيڑيا جا تا هے

میں زیادہ Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value