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# 8.7: Evaluating Exponential Expressions

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

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What if you had an exponential expression requiring multiple operations, like 2(14)2(14)3\begin{align*}2\left(\frac{1}{4}\right)^2 - \left(\frac{1}{4}\right)^3\end{align*}? How could you simplify it? After completing this Concept, you'll be able to use the order of operations to evaluate exponential expressions like this one.

### Guidance

When evaluating expressions we must keep in mind the order of operations. You must remember PEMDAS:

1. Evaluate inside the Parentheses.
2. Evaluate Exponents.
3. Perform Multiplication and Division operations from left to right.
4. Perform Addition and Subtraction operations from left to right.

#### Example A

Evaluate the following expressions.

a) 50\begin{align*}5^0\end{align*}

b) (23)3\begin{align*}\left(\frac{2}{3}\right)^3\end{align*}

c) 1612\begin{align*}16^{\frac{1}{2}}\end{align*}

d) 813\begin{align*}8^{-\frac{1}{3}}\end{align*}

Solution

a) 50=1\begin{align*}5^0=1\end{align*} A number raised to the power 0 is always 1.

b) (23)3=2333=827\begin{align*}\left(\frac{2}{3}\right)^3=\frac{2^3}{3^3}=\frac{8}{27}\end{align*}

c) 1612=16=4\begin{align*}16^{\frac{1}{2}}=\sqrt{16}=4\end{align*} Remember that an exponent of 12\begin{align*}\frac{1}{2}\end{align*} means taking the square root.

d) 813=1813=183=12\begin{align*}8^{-\frac{1}{3}}=\frac{1}{8^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{8}}=\frac{1}{2}\end{align*} Remember that an exponent of 13\begin{align*}\frac{1}{3}\end{align*} means taking the cube root.

#### Example B

Evaluate the following expressions.

a) 352105+1\begin{align*}3 \cdot 5^2-10 \cdot 5+1\end{align*}

b) 2423523222\begin{align*}\frac{2 \cdot 4^2-3 \cdot 5^2}{3^2-2^2}\end{align*}

c) (3322)234\begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}\end{align*}

Solution

a) Evaluate the exponent: 352105+1=325105+1\begin{align*}3 \cdot 5^2 - 10 \cdot 5+1=3 \cdot 25-10 \cdot 5+1\end{align*}

Perform multiplications from left to right: 325105+1=7550+1\begin{align*}3 \cdot 25-10 \cdot 5+1=75-50+1\end{align*}

Perform additions and subtractions from left to right: 7550+1=26\begin{align*}75-50+1=26\end{align*}

b) Treat the expressions in the numerator and denominator of the fraction like they are in parentheses: (242352)(3222)=(216325)(94)=(3275)5=435\begin{align*}\frac{(2 \cdot 4^2-3 \cdot 5^2)}{(3^2-2^2)}=\frac{(2 \cdot 16-3 \cdot 25)}{(9-4)}=\frac{(32-75)}{5}=\frac{-43}{5}\end{align*}

c) (3322)234=(2233)234=243634=2436322=2235=4243\begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}=\left(\frac{2^2}{3^3}\right)^2 \cdot \frac{3}{4}=\frac{2^4}{3^6} \cdot \frac{3}{4}=\frac{2^4}{3^6} \cdot \frac{3}{2^2}=\frac{2^2}{3^5}=\frac{4}{243}\end{align*}

#### Example C

Evaluate the following expressions for x=2,y=1,z=3\begin{align*}x = 2, y = - 1, z = 3\end{align*}.

a) 2x23y3+4z\begin{align*}2x^2-3y^3+4z\end{align*}

b) (x2y2)2\begin{align*}(x^2-y^2)^2\end{align*}

c) (3x2y54z)2\begin{align*}\left(\frac{3x^2y^5}{4z}\right)^{-2}\end{align*}

Solution

a) 2x23y3+4z=2223(1)3+43=243(1)+43=8+3+12=23\begin{align*}2x^2-3y^3+4z&=2 \cdot 2^2-3 \cdot (-1)^3+4 \cdot 3\\ &=2 \cdot 4-3 \cdot (-1)+4 \cdot 3=8+3+12\\ &=23\end{align*}

b) (x2y2)2=(22(1)2)2=(41)2=32=9\begin{align*}(x^2-y^2)^2=(2^2 - (-1)^2)^2=(4-1)^2=3^2=9\end{align*}

c)

\begin{align*}\left( \frac{3x^2y^5}{4z}\right)^{-2}&=\left( \frac{3 \cdot 2^2 \cdot (-1)^5}{4 \cdot 3}\right)^{-2}\\ &=\left(\frac{3 \cdot 4 \cdot (-1)}{12}\right)^{-2}\\ &=\left(\frac{-12}{12}\right)^{-2}\\ &=\left(\frac{-1}{1}\right)^{-2}\\ &=\left(\frac{1}{-1}\right)^2\\ &=(-1)^2\\ &=1\end{align*}

Watch this video for help with the Examples above.

### Vocabulary

• When evaluating expressions we must keep in mind the order of operations. You must remember PEMDAS:
1. Evaluate inside the Parentheses.
2. Evaluate Exponents.
3. Perform Multiplication and Division operations from left to right.
4. Perform Addition and Subtraction operations from left to right.

### Guided Practice

Evaluate the following expression for \begin{align*}x = 3, y = -2, z = -1\end{align*}.

\begin{align*}2z((x+1)^\frac{1}{2}-y^3)^2\end{align*}

Solution:

\begin{align*}2z((x+1)^\frac{1}{2}-y^3)^2&=2(-1)(((3)+1)^\frac{1}{2}-(-2)^3)^2\\ &=-2(4^\frac{1}{2}+8)^2\\ &=-2(2+8)^2\\ &=-2(10)^2\\ &=-200\end{align*}

### Practice

Evaluate the following expressions to a single number.

1. \begin{align*}3^{-2}\end{align*}
2. \begin{align*}{-4}^{-3}\end{align*}
3. \begin{align*}(6.2)^0\end{align*}
4. \begin{align*}8^{-4} \cdot 8^6\end{align*}
5. \begin{align*}\left (16^\frac{1}{2} \right )^3\end{align*}
6. \begin{align*}x^2 \cdot 4x^3 \cdot y^4 \cdot 4y^2\end{align*}, if \begin{align*}x = 2\end{align*} and \begin{align*}y = -1\end{align*}
7. \begin{align*}a^4(b^2)^3 + 2ab\end{align*}, if \begin{align*}a = -2\end{align*} and \begin{align*}b = 1\end{align*}
8. \begin{align*}5x^2 - 2y^3 + 3z\end{align*}, if \begin{align*}x = 3, y = 2,\end{align*} and \begin{align*}z = 4\end{align*}
9. \begin{align*}\left ( \frac{a^2}{b^3} \right )^{-2}\end{align*}, if \begin{align*}a = 5\end{align*} and \begin{align*}b = 3\end{align*}
10. \begin{align*}\left ( \frac{x^{-2}}{y^4} \right )^\frac{1}{2}\end{align*}, if \begin{align*}x=-3\end{align*} and \begin{align*}y=2\end{align*}

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