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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*}

Review (Answers)

To view the Review answers, open this PDF file and look for section 8.7. 

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