# Evaluating Exponential Expressions

## Evaluate numbers raised to positive, negative, and fractional powers

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Evaluating Exponential Expressions

### Evaluating Exponential Expressions

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.

#### Evaluating Expressions

1. Evaluate the following expressions.

a) \begin{align*}5^0\end{align*}

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

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

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

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

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

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

\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 \begin{align*}\frac{1}{3}\end{align*} means taking the cube root.

2. Evaluate the following expressions.

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

Evaluate the exponent: \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: \begin{align*}3 \cdot 25-10 \cdot 5+1=75-50+1\end{align*}

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

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

Treat the expressions in the numerator and denominator of the fraction like they are in parentheses: \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) \begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}\end{align*}

\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*}

#### Evaluating Expressions for Given Values

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

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

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

\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}\end{align*}

\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*}

### Example

#### Example 1

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*}

\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*}

### Review

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*}