Algebra 1 Worksheets | Exponents Worksheets - Free Printable

Problem: Evaluate each exponential function at the given value. Round to the nearest hundredth if needed.

We will solve each problem step by step.

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#### 1. \( g(n) = 3 \cdot \left( \frac{7}{6} \right)^n \) at \( n = -3 \)

\[
g(-3) = 3 \cdot \left( \frac{7}{6} \right)^{-3}
\]

First, calculate \( \left( \frac{7}{6} \right)^{-3} \):

\[
\left( \frac{7}{6} \right)^{-3} = \left( \frac{6}{7} \right)^3 = \frac{6^3}{7^3} = \frac{216}{343}
\]

Now, multiply by 3:

\[
g(-3) = 3 \cdot \frac{216}{343} = \frac{648}{343} \approx 1.89
\]

Answer: \( g(-3) \approx 1.89 \)

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#### 2. \( h(x) = 9 \cdot \left( \frac{1}{2} \right)^x \) at \( x = 3 \)

\[
h(3) = 9 \cdot \left( \frac{1}{2} \right)^3
\]

First, calculate \( \left( \frac{1}{2} \right)^3 \):

\[
\left( \frac{1}{2} \right)^3 = \frac{1}{8}
\]

Now, multiply by 9:

\[
h(3) = 9 \cdot \frac{1}{8} = \frac{9}{8} = 1.125
\]

Answer: \( h(3) = 1.13 \)

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#### 3. \( f(n) = \frac{4}{7} \cdot \left( \frac{1}{2} \right)^n \) at \( n = -3 \)

\[
f(-3) = \frac{4}{7} \cdot \left( \frac{1}{2} \right)^{-3}
\]

First, calculate \( \left( \frac{1}{2} \right)^{-3} \):

\[
\left( \frac{1}{2} \right)^{-3} = 2^3 = 8
\]

Now, multiply by \( \frac{4}{7} \):

\[
f(-3) = \frac{4}{7} \cdot 8 = \frac{32}{7} \approx 4.57
\]

Answer: \( f(-3) \approx 4.57 \)

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#### 4. \( h(n) = \frac{1}{7} \cdot 2^n \) at \( n = 2 \)

\[
h(2) = \frac{1}{7} \cdot 2^2
\]

First, calculate \( 2^2 \):

\[
2^2 = 4
\]

Now, multiply by \( \frac{1}{7} \):

\[
h(2) = \frac{1}{7} \cdot 4 = \frac{4}{7} \approx 0.57
\]

Answer: \( h(2) \approx 0.57 \)

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#### 5. \( g(y) = \frac{9}{3} \cdot \left( \frac{1}{2} \right)^y \) at \( y = 2 \)

Simplify \( \frac{9}{3} \):

\[
\frac{9}{3} = 3
\]

So, the function becomes:

\[
g(y) = 3 \cdot \left( \frac{1}{2} \right)^y
\]

At \( y = 2 \):

\[
g(2) = 3 \cdot \left( \frac{1}{2} \right)^2
\]

Calculate \( \left( \frac{1}{2} \right)^2 \):

\[
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]

Now, multiply by 3:

\[
g(2) = 3 \cdot \frac{1}{4} = \frac{3}{4} = 0.75
\]

Answer: \( g(2) = 0.75 \)

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#### 6. \( h(x) = 5 \cdot 2^x \) at \( x = 3 \)

\[
h(3) = 5 \cdot 2^3
\]

Calculate \( 2^3 \):

\[
2^3 = 8
\]

Now, multiply by 5:

\[
h(3) = 5 \cdot 8 = 40
\]

Answer: \( h(3) = 40.00 \)

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#### 7. \( f(x) = 3 \cdot \left( \frac{5}{7} \right)^x \) at \( x = 3 \)

\[
f(3) = 3 \cdot \left( \frac{5}{7} \right)^3
\]

First, calculate \( \left( \frac{5}{7} \right)^3 \):

\[
\left( \frac{5}{7} \right)^3 = \frac{5^3}{7^3} = \frac{125}{343}
\]

Now, multiply by 3:

\[
f(3) = 3 \cdot \frac{125}{343} = \frac{375}{343} \approx 1.10
\]

Answer: \( f(3) \approx 1.10 \)

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#### 8. \( h(n) = \frac{1}{2} \cdot \left( \frac{1}{3} \right)^n \) at \( n = -2 \)

\[
h(-2) = \frac{1}{2} \cdot \left( \frac{1}{3} \right)^{-2}
\]

First, calculate \( \left( \frac{1}{3} \right)^{-2} \):

\[
\left( \frac{1}{3} \right)^{-2} = 3^2 = 9
\]

Now, multiply by \( \frac{1}{2} \):

\[
h(-2) = \frac{1}{2} \cdot 9 = \frac{9}{2} = 4.5
\]

Answer: \( h(-2) = 4.50 \)

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#### 9. \( g(y) = \frac{2}{7} \cdot 2^y \) at \( y = 2 \)

\[
g(2) = \frac{2}{7} \cdot 2^2
\]

Calculate \( 2^2 \):

\[
2^2 = 4
\]

Now, multiply by \( \frac{2}{7} \):

\[
g(2) = \frac{2}{7} \cdot 4 = \frac{8}{7} \approx 1.14
\]

Answer: \( g(2) \approx 1.14 \)

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#### 10. \( f(y) = \frac{3}{2} \cdot 2^y \) at \( y = -2 \)

\[
f(-2) = \frac{3}{2} \cdot 2^{-2}
\]

First, calculate \( 2^{-2} \):

\[
2^{-2} = \frac{1}{2^2} = \frac{1}{4}
\]

Now, multiply by \( \frac{3}{2} \):

\[
f(-2) = \frac{3}{2} \cdot \frac{1}{4} = \frac{3}{8} = 0.375
\]

Answer: \( f(-2) = 0.38 \)

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#### 11. \( f(x) = 4 \cdot 2^x \) at \( x = -2 \)

\[
f(-2) = 4 \cdot 2^{-2}
\]

First, calculate \( 2^{-2} \):

\[
2^{-2} = \frac{1}{2^2} = \frac{1}{4}
\]

Now, multiply by 4:

\[
f(-2) = 4 \cdot \frac{1}{4} = 1
\]

Answer: \( f(-2) = 1.00 \)

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#### 12. \( h(y) = \frac{1}{2} \cdot \left( \frac{1}{3} \right)^y \) at \( y = 3 \)

\[
h(3) = \frac{1}{2} \cdot \left( \frac{1}{3} \right)^3
\]

First, calculate \( \left( \frac{1}{3} \right)^3 \):

\[
\left( \frac{1}{3} \right)^3 = \frac{1}{27}
\]

Now, multiply by \( \frac{1}{2} \):

\[
h(3) = \frac{1}{2} \cdot \frac{1}{27} = \frac{1}{54} \approx 0.02
\]

Answer: \( h(3) \approx 0.02 \)

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#### 13. \( g(y) = \frac{1}{2} \cdot \left( \frac{2}{3} \right)^y \) at \( y = -2 \)

\[
g(-2) = \frac{1}{2} \cdot \left( \frac{2}{3} \right)^{-2}
\]

First, calculate \( \left( \frac{2}{3} \right)^{-2} \):

\[
\left( \frac{2}{3} \right)^{-2} = \left( \frac{3}{2} \right)^2 = \frac{3^2}{2^2} = \frac{9}{4}
\]

Now, multiply by \( \frac{1}{2} \):

\[
g(-2) = \frac{1}{2} \cdot \frac{9}{4} = \frac{9}{8} = 1.125
\]

Answer: \( g(-2) = 1.13 \)

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#### 14. \( g(y) = 8 \cdot \left( \frac{1}{2} \right)^y \) at \( y = -2 \)

\[
g(-2) = 8 \cdot \left( \frac{1}{2} \right)^{-2}
\]

First, calculate \( \left( \frac{1}{2} \right)^{-2} \):

\[
\left( \frac{1}{2} \right)^{-2} = 2^2 = 4
\]

Now, multiply by 8:

\[
g(-2) = 8 \cdot 4 = 32
\]

Answer: \( g(-2) = 32.00 \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 1.89 \\
2. & 1.13 \\
3. & 4.57 \\
4. & 0.57 \\
5. & 0.75 \\
6. & 40.00 \\
7. & 1.10 \\
8. & 4.50 \\
9. & 1.14 \\
10. & 0.38 \\
11. & 1.00 \\
12. & 0.02 \\
13. & 1.13 \\
14. & 32.00 \\
\end{array}
}
\]