Powers and Exponents - Free Printable

Let's solve each of these exponential function problems step by step. We'll evaluate each function at the given value and round to the nearest hundredth if needed.

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

$$
g(-3) = 3 \cdot \left(\frac{7}{6}\right)^{-3} = 3 \cdot \left(\frac{6}{7}\right)^3 = 3 \cdot \frac{216}{343} = \frac{648}{343} \approx 1.89
$$

✔ Answer: 1.89

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

$$
h(3) = 9 \cdot \left(\frac{1}{2}\right)^3 = 9 \cdot \frac{1}{8} = \frac{9}{8} = 1.125 \approx 1.13
$$

✔ Answer: 1.13

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

$$
f(-3) = \frac{4}{7} \cdot \left(\frac{1}{2}\right)^{-3} = \frac{4}{7} \cdot 2^3 = \frac{4}{7} \cdot 8 = \frac{32}{7} \approx 4.57
$$

✔ Answer: 4.57

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

$$
h(2) = \frac{1}{7} \cdot 2^2 = \frac{1}{7} \cdot 4 = \frac{4}{7} \approx 0.57
$$

✔ Answer: 0.57

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

Note: $ \frac{9}{3} = 3 $

$$
g(2) = 3 \cdot \left(\frac{1}{2}\right)^2 = 3 \cdot \frac{1}{4} = \frac{3}{4} = 0.75
$$

✔ Answer: 0.75

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

$$
h(3) = 5 \cdot 2^3 = 5 \cdot 8 = 40
$$

✔ Answer: 40

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7) $ f(x) = 3 \cdot \left(\frac{9}{7}\right)^x $ at $ x = 3 $

$$
f(3) = 3 \cdot \left(\frac{9}{7}\right)^3 = 3 \cdot \frac{729}{343} = \frac{2187}{343} \approx 6.37
$$

✔ Answer: 6.37

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

$$
h(-2) = \frac{1}{2} \cdot \left(\frac{1}{3}\right)^{-2} = \frac{1}{2} \cdot 3^2 = \frac{1}{2} \cdot 9 = \frac{9}{2} = 4.5
$$

✔ Answer: 4.50

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

$$
g(2) = \frac{2}{7} \cdot 2^2 = \frac{2}{7} \cdot 4 = \frac{8}{7} \approx 1.14
$$

✔ Answer: 1.14

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

$$
f(-2) = \frac{3}{2} \cdot 2^{-2} = \frac{3}{2} \cdot \frac{1}{4} = \frac{3}{8} = 0.375 \approx 0.38
$$

✔ Answer: 0.38

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

$$
f(-2) = 4 \cdot 2^{-2} = 4 \cdot \frac{1}{4} = 1
$$

✔ Answer: 1.00

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

$$
h(3) = \frac{1}{2} \cdot \left(\frac{1}{3}\right)^3 = \frac{1}{2} \cdot \frac{1}{27} = \frac{1}{54} \approx 0.0185 \approx 0.02
$$

✔ Answer: 0.02

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

$$
g(-2) = \frac{1}{2} \cdot \left(\frac{2}{3}\right)^{-2} = \frac{1}{2} \cdot \left(\frac{3}{2}\right)^2 = \frac{1}{2} \cdot \frac{9}{4} = \frac{9}{8} = 1.125 \approx 1.13
$$

✔ Answer: 1.13

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

$$
g(-2) = 8 \cdot \left(\frac{1}{2}\right)^{-2} = 8 \cdot 2^2 = 8 \cdot 4 = 32
$$

✔ Answer: 32

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | 1.89 |
| 2) | 1.13 |
| 3) | 4.57 |
| 4) | 0.57 |
| 5) | 0.75 |
| 6) | 40 |
| 7) | 6.37 |
| 8) | 4.50 |
| 9) | 1.14 |
| 10) | 0.38 |
| 11) | 1.00 |
| 12) | 0.02 |
| 13) | 1.13 |
| 14) | 32 |

Let me know if you'd like this formatted as a printable answer sheet!