<meta http-equiv="refresh" content="1; url=/nojavascript/">

# 6.6: Exponential Expressions

Difficulty Level: Advanced Created by: CK-12
%
Progress
Practice Evaluating Exponential Expressions
Progress
%

Can you simplify the following expression so that it has only positive exponents?

$\frac{8x^3y^{-2}}{(-4a^2b^4)^{-2}}$

### Guidance

The following table summarizes all of the rules for exponents.

Laws of Exponents

If $a \in R, a \ge 0$ and $m, n \in Q$ , then

1. $a^m \times a^n=a^{m+n}$
2. $\frac{a^m}{a^n}=a^{m-n} \ (\text{if} \ m > n, a \neq 0)$
3. $(a^m)^n=a^{mn}$
4. $(ab)^n=a^nb^n$
5. $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \ (b \neq 0)$
6. $a^0=1 \ (a \neq 0)$
7. $a^{-m}=\frac{1}{a^m}$
8. $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m$

#### Example A

Evaluate $81^{-\frac{1}{4}}$ .

Solution: First, rewrite with a positive exponent:

$81^{-\frac{1}{4}}=\frac{1}{81^{\frac{1}{4}}}=\left(\frac{1}{81}\right)^{\frac{1}{4}}$ .

Next, evaluate the fractional exponent:

$\left(\frac{1}{81}\right)^{\frac{1}{4}}=\sqrt[4]{\frac{1}{81}}=\frac{1}{3}$

#### Example B

Simplify $(4x^3 y) (3x^5 y^2 )^4$ .

Solution:

$(4x^3 y) (3x^5 y^2 )^4&=(4x^3 y) (81x^{20} y^8 )\\ & =324x^{23}y^9$

#### Example C

Simplify $\left(\frac{x^{-2}y}{x^4y^3}\right)^{-2}$ .

Solution:

$\left(\frac{x^{-2}y}{x^4y^3}\right)^{-2}&=\left(\frac{x^4y^3}{x^{-2}y}\right)^{2}\\ &=(x^6y^2)^{2}\\ &=x^{12}y^4$

#### Concept Problem Revisited

$\frac{8x^3y^{-2}}{(-4x^2y^4)^{-2}}&=(8x^3y^{-2})(-4x^2y^4)^2\\&=(8x^3y^{-2})(16x^4y^8) \\ &=8\cdot 16 \cdot x^3 \cdot x^4 \cdot y^{-2} \cdot y^8\\ &=128x^7y^6$

### Guided Practice

Use the laws of exponents to simplify each of the following:

1. $(-2x)^5 (2x^2)$

2. $(16x^{10}) \left(\frac{3}{4}x^5\right)$

3. $\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}$

1. $(-2x)^5 (2x^2)=(-32x^5)(2x^2)=-64x^7$

2. $(16x^{10}) \left(\frac{3}{4}x^5\right)=12x^{15}$

3. $\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}=\frac{x^{64}}{x^{56}}=x^8$

### Explore More

Simplify each expression.

1. $(x^{10}) (x^{10})$
2. $(7x^3)(3x^7)$
3. $(x^3 y^2) (xy^3) (x^5 y)$
4. $\frac{(x^3)(x^2)}{(x^4)}$
5. $\frac{x^2}{x^{-3}}$
6. $\frac{x^6 y^8}{x^4 y^{-2}}$
7. $(2x^{12})^3$
8. $(x^5 y^{10})^7$
9. $\left(\frac{2x^{10}}{3y^{20}}\right)^3$

Express each of the following as a power of 3. Do not evaluate.

1. $(3^3)^5$
2. $(3^9)(3^3)$
3. $(9)(3^7)$
4. $9^4$
5. $(9)(27^2)$

Apply the laws of exponents to evaluate each of the following without using a calculator.

1. $(2^3)(2^2)$
2. $6^6 \div 6^5$
3. $-(3^2)^3$
4. $(1^2)^3+(1^3)^2$
5. $\left(\frac{1}{3}\right)^6 \div \left(\frac{1}{3}\right)^8$

Use the laws of exponents to simplify each of the following.

1. $(4x)^2$
2. $(-3x)^3$
3. $(x^3)^4$
4. $(3x)(x^7)$
5. $(5x)(4x^4)$
6. $(-3x^2)(-6x^3)$
7. $(10x^8) \div (2x^4)$

Simplify each of the following using the laws of exponents.

1. $5^{\frac{1}{2}} \times 5^{\frac{1}{3}}$
2. $(d^4 e^8 f^{12})^{\frac{1}{4}}$
3. $\sqrt[4]{\frac{y^{\frac{1}{2}} \sqrt{xy}}{x^{\frac{2}{3}}}}$
4. $(32a^{20}b^{-15})^{\frac{1}{5}}$
5. $(729x^{12}y^{-6})^{\frac{2}{3}}$

## Date Created:

Apr 30, 2013

Feb 26, 2015
You can only attach files to Modality which belong to you
If you would like to associate files with this Modality, please make a copy first.