Students can use this worksheet to practice substituting values into exponential equations involving fractions and whole numbers.

Let’s solve each problem one by one. We’ll plug in the given value for the variable and calculate step by step. Remember: when you see a negative exponent, it means “take the reciprocal” — like \( a^{-n} = \frac{1}{a^n} \). And we’ll round to the nearest hundredth if needed.

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1) g(n) = 3 • (5/6)^n at n = -3

First, plug in n = -3:

g(-3) = 3 • (5/6)^{-3}

Negative exponent → flip the fraction:

= 3 • (6/5)^3

Now compute (6/5)^3 = 6³ / 5³ = 216 / 125 = 1.728

Then multiply by 3:

3 × 1.728 = 5.184

Rounded to nearest hundredth: 5.18

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2) h(x) = 9 • (1/2)^x at x = 3

h(3) = 9 • (1/2)^3 = 9 • (1/8) = 9/8 = 1.125

Rounded to nearest hundredth: 1.13

Wait — 1.125 is exactly halfway between 1.12 and 1.13? Actually, standard rounding: look at thousandths place — it’s 5, so round up.

→ 1.13

But let me double-check: 9 ÷ 8 = 1.125 → yes, rounds to 1.13

Actually, wait — some systems round 0.5 up, others use "round half to even". But for school math, usually round 0.5 up.

So 1.125 → 1.13

BUT — hold on! Let me check again: 1.125 to nearest hundredth — the digit in thousandths is 5, so we round up the hundredths place (which is 2) to 3 → 1.13. Correct.

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3) f(n) = (4/7) • (1/2)^n at n = -3

f(-3) = (4/7) • (1/2)^{-3} = (4/7) • (2^3) = (4/7) • 8 = 32/7 ≈ 4.5714...

Rounded to nearest hundredth: 4.57

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4) h(n) = (1/7) • 2^n at n = 2

h(2) = (1/7) • 2² = (1/7) • 4 = 4/7 ≈ 0.5714...

Rounded to nearest hundredth: 0.57

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5) g(y) = (9/3) • (1/2)^y at y = 2

First, simplify 9/3 = 3

g(2) = 3 • (1/2)^2 = 3 • (1/4) = 3/4 = 0.75

No rounding needed.

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6) h(x) = 5 • 2^x at x = 3

h(3) = 5 • 2³ = 5 • 8 = 40

Exact number.

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7) f(x) = 3 • (5/7)^x at x = 3

f(3) = 3 • (5/7)^3 = 3 • (125 / 343)

Compute 125 ÷ 343 ≈ 0.364431...

Multiply by 3: ≈ 1.09329...

Rounded to nearest hundredth: 1.09

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8) h(n) = (1/2) • (1/3)^n at n = -2

h(-2) = (1/2) • (1/3)^{-2} = (1/2) • (3^2) = (1/2) • 9 = 9/2 = 4.5

Or 4.50 if writing to hundredths.

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9) g(y) = (2/7) • 2^y at y = 2

g(2) = (2/7) • 2² = (2/7) • 4 = 8/7 ≈ 1.142857...

Rounded to nearest hundredth: 1.14

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10) f(y) = (3/2) • 2^y at y = -2

f(-2) = (3/2) • 2^{-2} = (3/2) • (1/4) = 3/8 = 0.375

Rounded to nearest hundredth: 0.38

(Because 0.375 → thousandths digit is 5 → round up → 0.38)

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11) f(x) = 4 • 2^x at x = -2

f(-2) = 4 • 2^{-2} = 4 • (1/4) = 1

Exact.

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12) h(y) = (1/2) • (1/3)^y at y = 3

h(3) = (1/2) • (1/3)^3 = (1/2) • (1/27) = 1/54 ≈ 0.018518...

Rounded to nearest hundredth: 0.02

(Since 0.0185... → hundredths place is 1, thousandths is 8 → round up → 0.02)

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13) g(y) = (1/2) • (2/3)^y at y = -2

g(-2) = (1/2) • (2/3)^{-2} = (1/2) • (3/2)^2 = (1/2) • (9/4) = 9/8 = 1.125

Rounded to nearest hundredth: 1.13

(Same as #2 — 1.125 → 1.13)

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14) g(y) = 8 • (1/2)^y at y = -2

g(-2) = 8 • (1/2)^{-2} = 8 • (2^2) = 8 • 4 = 32

Exact.

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Now let’s list all answers clearly:

1) 5.18
2) 1.13
3) 4.57
4) 0.57
5) 0.75
6) 40
7) 1.09
8) 4.50
9) 1.14
10) 0.38
11) 1
12) 0.02
13) 1.13
14) 32

Double-checked calculations — especially rounding steps.

Final Answer:
1) 5.18
2) 1.13
3) 4.57
4) 0.57
5) 0.75
6) 40
7) 1.09
8) 4.50
9) 1.14
10) 0.38
11) 1
12) 0.02
13) 1.13
14) 32