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Use Multiplication Properties of Exponents

Simplify expressions with 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 4th power. In part ⓑ we raise just the 5 to the 4th 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.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Simplify: \(\frac{3}{4}\cdot \frac{3}{4}.\)
    If you missed this problem, review .

    Rivela la risposta

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

  2. Simplify: \((-2)(-2)(-2).\)
    If you missed this problem, review .

    Rivela la risposta

    \(-8\)

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

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

    Rivela la risposta

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

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

    Rivela la risposta

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

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

    Rivela la risposta
    ⓐ \({(-5)}^{4}\)
    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: ⓐ \({(-3)}^{4}\) ⓑ \(\text{-}{3}^{4}.\)

    Rivela la risposta

    ⓐ \(81\) ⓑ \(-81\)

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

    Rivela la risposta

    ⓐ \(169\) ⓑ \(-169\)

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

    Rivela la risposta

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

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

    Rivela la risposta

    \({b}^{17}\)

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

    Rivela la risposta

    \({x}^{16}\)

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

    Rivela la risposta
    1. ⓐ
      Use the product property, am · an = am+n.
      Simplify.
    2. ⓑ
      Use the product property, am · an = am+n.
      Simplify.

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

    Rivela la risposta

    ⓐ \({5}^{6}\) ⓑ \({4}^{18}\)

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

    Rivela la risposta

    ⓐ \({7}^{14}\) ⓑ \({10}^{11}\)

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

    Rivela la risposta
    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: ⓐ \({p}^{5}\cdot p\) ⓑ \({y}^{14}\cdot {y}^{29}.\)

    Rivela la risposta

    ⓐ \({p}^{6}\) ⓑ \({y}^{43}\)

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

    Rivela la risposta

    ⓐ \({z}^{8}\) ⓑ \({b}^{49}\)

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

    Rivela la risposta

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

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

    Rivela la risposta

    \({x}^{18}\)

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

    Rivela la risposta

    \({b}^{19}\)

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

    Rivela la risposta

    ⓐ

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


    ⓑ
    Use the power property.
    Simplify.

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

    Rivela la risposta

    ⓐ \({b}^{35}\) ⓑ \({5}^{12}\)

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

    Rivela la risposta

    ⓐ \({z}^{54}\) ⓑ \({3}^{49}\)

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

    Rivela la risposta
    1. ⓐ
      Use Power of a Product Property, (ab)m = ambm.
      Simplify.
    2. ⓑ
      Use Power of a Product Property, (ab)m = ambm.
      Simplify.

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

    Rivela la risposta

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

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

    Rivela la risposta

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

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

    Rivela la risposta
    ⓐ \({({y}^{3})}^{6}{({y}^{5})}^{4}\)
    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: ⓐ \({({a}^{4})}^{5}{({a}^{7})}^{4}\) ⓑ \({(-2{c}^{4}{d}^{2})}^{3}.\)

    Rivela la risposta

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

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

    Rivela la risposta

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

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

    Rivela la risposta
    ⓐ \({(5m)}^{2}(3{m}^{3})\)
    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: ⓐ \({(5n)}^{2}(3{n}^{10})\) ⓑ \({({c}^{4}{d}^{2})}^{5}{(3c{d}^{5})}^{4}.\)

    Rivela la risposta

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

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

    Rivela la risposta

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

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

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

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

    Rivela la risposta

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

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

    Rivela la risposta

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

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

    Rivela la risposta
    \((\frac{5}{6}{x}^{3}y)(12x{y}^{2})\)
    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: \((\frac{2}{5}{a}^{4}{b}^{3})(15a{b}^{3}).\)

    Rivela la risposta

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

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

    Rivela la risposta

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

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

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

    Rivela la risposta

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

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{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.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Use Multiplication Properties of Exponents

  1. Simplify expressions with exponents
  2. Simplify expressions using the Product Property for Exponents
  3. Simplify expressions using the Power Property for Exponents
  4. Simplify expressions using the Product to a Power Property
  5. Simplify expressions by applying several properties
  6. Multiply monomials
  7. If

Questions people ask

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.

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Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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