# 6.6: Exponential Expressions

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

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

\begin{align*}\frac{8x^3y^{-2}}{(-4a^2b^4)^{-2}}\end{align*}

### Exponential Expressions

Here is a summary of all the Laws of Exponents that have been covered so far:

Laws of Exponents

If \begin{align*}a \in R, a \ge 0\end{align*} and \begin{align*}m, n \in Q\end{align*}, then

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

#### Now, applying the Laws of Exponents, let's evaluate and simplify the following expressions:

1. \begin{align*}81^{-\frac{1}{4}}\end{align*}

First, rewrite with a positive exponent:

\begin{align*}81^{-\frac{1}{4}}=\frac{1}{81^{\frac{1}{4}}}=\left(\frac{1}{81}\right)^{\frac{1}{4}}\end{align*}.

Next, evaluate the fractional exponent:

\begin{align*}\left(\frac{1}{81}\right)^{\frac{1}{4}}=\sqrt[4]{\frac{1}{81}}=\frac{1}{3}\end{align*}

1. \begin{align*}(4x^3 y) (3x^5 y^2 )^4 \end{align*}

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

1. \begin{align*}\left(\frac{x^{-2}y}{x^4y^3}\right)^{-2}\end{align*}

\begin{align*}\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\end{align*}

### Examples

#### Example 1

Earlier, you were asked to simplify the following expression:

\begin{align*}\frac{8x^3y^{-2}}{(-4a^2b^4)^{-2}}\end{align*}

\begin{align*}\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\end{align*}

#### Example 2

Use the Laws of Exponents to simplify the following:

\begin{align*}(-2x)^5 (2x^2)\end{align*}

\begin{align*}(-2x)^5 (2x^2)=(-32x^5)(2x^2)=-64x^7\end{align*}

#### Example 3

Use the Laws of Exponents to simplify the following:

\begin{align*}(16x^{10}) \left(\frac{3}{4}x^5\right)\end{align*}

\begin{align*}(16x^{10}) \left(\frac{3}{4}x^5\right)=12x^{15}\end{align*}

#### Example 4

Use the Laws of Exponents to simplify the following:

\begin{align*}\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}\end{align*}

\begin{align*}\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}=\frac{x^{64}}{x^{56}}=x^8\end{align*}

### Review

Simplify each expression.

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

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

1. \begin{align*}(3^3)^5\end{align*}
2. \begin{align*}(3^9)(3^3)\end{align*}
3. \begin{align*}(9)(3^7)\end{align*}
4. \begin{align*}9^4\end{align*}
5. \begin{align*}(9)(27^2)\end{align*}

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

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

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

1. \begin{align*}(4x)^2\end{align*}
2. \begin{align*}(-3x)^3\end{align*}
3. \begin{align*}(x^3)^4\end{align*}
4. \begin{align*}(3x)(x^7)\end{align*}
5. \begin{align*}(5x)(4x^4)\end{align*}
6. \begin{align*}(-3x^2)(-6x^3)\end{align*}
7. \begin{align*}(10x^8) \div (2x^4)\end{align*}

Simplify each of the following using the laws of exponents.

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

To see the Review answers, open this PDF file and look for section 6.6.

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