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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 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:

ⓐ \(\ {5}^{3}\)
ⓑ \(\ {9}^{1}\)

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:

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

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:

ⓐ \(\ {(-3)}^{4}\)
ⓑ \(\ {-3}^{4}\)

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:

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

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.

\({4}^{5}\)

Solution

1,024

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.

Practice (40)

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

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

Odhaliť odpoveď

\(\frac{9}{16}\)
Simplify: \((-2)(-2)(-2).\)
If you missed the problem, review .

Odhaliť odpoveď

\(-8\)
Simplify:
1. ⓐ \(\ {5}^{3}\)
2. ⓑ \(\ {9}^{1}\)
Odhaliť odpoveď

ⓐ
\({5}^{3}\)
Multiply 3 factors of 5. \(5\cdot 5\cdot 5\)
Simplify. \(125\)
ⓑ
\({9}^{1}\)
Multiply 1 factor of 9. \(9\)
Simplify:
1. ⓐ \(\ {4}^{3}\)
2. ⓑ \(\ {11}^{1}\)
Odhaliť odpoveď
1. ⓐ 64
2. ⓑ 11
Simplify:
1. ⓐ \(\ {3}^{4}\)
2. ⓑ \(\ {21}^{1}\)
Odhaliť odpoveď
1. ⓐ 81
2. ⓐ 21
Simplify:
1. ⓐ \(\ {(\frac{7}{8})}^{2}\)
2. ⓑ \(\ {(0.74)}^{2}\)
Odhaliť odpoveď

ⓐ
\({(\frac{7}{8})}^{2}\)
Multiply two factors. \((\frac{7}{8})(\frac{7}{8})\)
Simplify. \(\frac{49}{64}\)
ⓑ
\({(0.74)}^{2}\)
Multiply two factors. \((0.74)(0.74)\)
Simplify. \(0.5476\)
Simplify:
1. ⓐ \(\ {(\frac{5}{8})}^{2}\)
2. ⓑ \(\ {(0.67)}^{2}\)
Odhaliť odpoveď
1. ⓐ \(\ \frac{25}{64}\)
2. ⓑ \(\ 0.4489\)
Simplify:
1. ⓐ \(\ {(\frac{2}{5})}^{3}\)
2. ⓑ \(\ {(0.127)}^{2}\)
Odhaliť odpoveď
1. ⓐ \(\ \frac{8}{125}\)
2. ⓑ \(\ 0.016129\)
Simplify:
1. ⓐ \(\ {(-3)}^{4}\)
2. ⓑ \(\ {-3}^{4}\)
Odhaliť odpoveď

ⓐ
\({(-3)}^{4}\)
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:
1. ⓐ \(\ {(-2)}^{4}\)
2. ⓑ \(\ {-2}^{4}\)
Odhaliť odpoveď
1. ⓐ 16
2. ⓑ −16
Simplify:
1. ⓐ \(\ {(-8)}^{2}\)
2. ⓑ \(\ {-8}^{2}\)
Odhaliť odpoveď
1. ⓐ 64
2. ⓑ −64
Simplify: \({x}^{5}\cdot {x}^{7}.\)

Odhaliť odpoveď

\({x}^{5}\cdot {x}^{7}\)
Use the product property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify. \({x}^{12}\)
Simplify: \({x}^{7}\cdot {x}^{8}.\)

Odhaliť odpoveď

x¹⁵
Simplify: \({x}^{5}\cdot {x}^{11}.\)

Odhaliť odpoveď

x¹⁶
Simplify: \({b}^{4}\cdot b.\)

Odhaliť odpoveď

\({b}^{4}\cdot b\)
Rewrite, \(b={b}^{1}.\) \({b}^{4}\cdot {b}^{1}\)
Use the product property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify. \({b}^{5}\)
Simplify: \({p}^{9}\cdot p.\)

Odhaliť odpoveď

p¹⁰
Simplify: \(m\cdot {m}^{7}.\)

Odhaliť odpoveď

m⁸
Simplify: \({2}^{7}\cdot {2}^{9}.\)

Odhaliť odpoveď

\({2}^{7}\cdot {2}^{9}\)
Use the product property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify. \({2}^{16}\)
Simplify: \(6\cdot {6}^{9}.\)

Odhaliť odpoveď

6¹⁰
Simplify: \({9}^{6}\cdot {9}^{9}.\)

Odhaliť odpoveď

9¹⁵
Simplify: \({y}^{17}\cdot {y}^{23}.\)

Odhaliť odpoveď

\({y}^{17}\cdot {y}^{23}\)
Notice, the bases are the same, so add the exponents.
Simplify. \({y}^{40}\)
Simplify: \({y}^{24}\cdot {y}^{19}.\)

Odhaliť odpoveď

y⁴³
Simplify: \({z}^{15}\cdot {z}^{24}.\)

Odhaliť odpoveď

z³⁹
Simplify: \({x}^{3}\cdot {x}^{4}\cdot {x}^{2}.\)

Odhaliť odpoveď

\({x}^{3}\cdot {x}^{4}\cdot {x}^{2}\)
Add the exponents, since the bases are the same.
Simplify. \({x}^{9}\)
Simplify: \({x}^{7}\cdot {x}^{5}\cdot {x}^{9}.\)

Odhaliť odpoveď

x²¹
Simplify: \({y}^{3}\cdot {y}^{8}\cdot {y}^{4}.\)

Odhaliť odpoveď

y¹⁵
Simplify:
1. ⓐ \(\ {({x}^{5})}^{7}\)
2. ⓐ \(\ {({3}^{6})}^{8}\)
Odhaliť odpoveď

ⓐ
\({({x}^{5})}^{7}\)
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:
1. ⓐ \(\ {({x}^{7})}^{4}\)
2. ⓑ \(\ {({7}^{4})}^{8}\)
Odhaliť odpoveď
1. ⓐ x²⁸
2. ⓑ 7³²
Simplify:
1. ⓐ \(\ {({x}^{6})}^{9}\)
2. ⓑ \(\ {({8}^{6})}^{7}\)
Odhaliť odpoveď
1. ⓐ x⁵⁴
2. ⓑ 8⁴²
Simplify: \({(-11x)}^{2}.\)

Odhaliť odpoveď

\({(-11x)}^{2}\)
Use the Power of a Product Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)
Simplify. \(121{x}^{2}\)
Simplify: \({(-14x)}^{2}.\)

Odhaliť odpoveď

196x²
Simplify: \({(-12a)}^{2}.\)

Odhaliť odpoveď

144a²
Simplify: \({(3xy)}^{3}.\)

Odhaliť odpoveď

\({(3xy)}^{3}\)
Raise each factor to the third power.
Simplify. \(27{x}^{3}{y}^{3}\)
Simplify: \({(-4xy)}^{4}.\)

Odhaliť odpoveď

256x⁴y⁴
Simplify: \({(6xy)}^{3}.\)

Odhaliť odpoveď

216x³y³
Simplify: \({({x}^{2})}^{6}{({x}^{5})}^{4}.\)

Odhaliť odpoveď

\({({x}^{2})}^{6}{({x}^{5})}^{4}\)
Use the Power Property. \({x}^{12}\cdot {x}^{20}\)
Add the exponents. \({x}^{32}\)
Simplify: \({({x}^{4})}^{3}{({x}^{7})}^{4}.\)

Odhaliť odpoveď

x⁴⁰
Simplify: \({({y}^{9})}^{2}{({y}^{8})}^{3}.\)

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y⁴²
Simplify: \({(-7{x}^{3}{y}^{4})}^{2}.\)

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

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−512x¹²y²¹

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.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Use Multiplication Properties of Exponents

Simplify expressions with exponents
Simplify expressions using the Product Property of Exponents
Simplify expressions using the Power Property of Exponents
Simplify expressions using the Product to a Power Property
Simplify expressions by applying several properties
Multiply monomials
If
To multiply with like bases, add the exponents.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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

Viac v kategórii Arithmetic

Order of operations Fractions Percentages Powers and roots