Simplify exponents in algebraic quotients on this math worksheet.

Let’s go through each problem one by one. We’re simplifying expressions where a quotient (fraction) is raised to a power. The rule we use is:

> When you have (a/b)^n, it becomes a^n / b^n.
> Also, when you have (x^m)^n, it becomes x^(m×n).

We’ll apply these rules carefully.

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Problem 1:
(5d² / 8d⁴)³

Step 1: Simplify inside the parentheses first.
d² / d⁴ = d^(2-4) = d^(-2) → so we get 5/8 × d^(-2) = 5/(8d²)

But actually, better to keep as fraction and raise numerator and denominator separately.

So:
Numerator: (5d²)³ = 5³ × (d²)³ = 125 × d⁶
Denominator: (8d⁴)³ = 8³ × (d⁴)³ = 512 × d¹²

Now combine:
125d⁶ / 512d¹² = 125 / 512 × d^(6-12) = 125 / 512 × d^(-6) = 125 / (512d⁶)

✔ Final for #1: 125 / (512d⁶)

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Problem 2:
(c / c⁴)³

Inside: c / c⁴ = c^(1-4) = c^(-3)

Then raise to 3rd power: (c^(-3))³ = c^(-9) = 1 / c⁹

✔ Final for #2: 1 / c⁹

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Problem 3:
(5g³b⁸ / 4g²b³)³

Simplify inside first:

g³ / g² = g^(3-2) = g¹
b⁸ / b³ = b^(8-3) = b⁵
So inside becomes: (5/4) × g × b⁵

Now cube it:
(5/4)³ × g³ × (b⁵)³ = 125/64 × g³ × b¹⁵

✔ Final for #3: (125g³b¹⁵) / 64

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Problem 4:
(7r / 2r²)²

Inside: r / r² = r^(-1) → so 7/2 × r^(-1) = 7/(2r)

Square it: (7/(2r))² = 49 / (4r²)

Or step-by-step:

Numerator: (7r)² = 49r²
Denominator: (2r²)² = 4r⁴
→ 49r² / 4r⁴ = 49 / (4r²)

✔ Final for #4: 49 / (4r²)

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Problem 5:
(8dg² / 3d⁴g⁴)³

Inside:
d / d⁴ = d^(-3)
g² / g⁴ = g^(-2)
So: 8/3 × d^(-3) × g^(-2) = 8 / (3d³g²)

Cube it:
8³ / [3³ × (d³)³ × (g²)³] = 512 / (27 × d⁹ × g⁶)

✔ Final for #5: 512 / (27d⁹g⁶)

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Problem 6:
(8w² / 3w²k⁴)²

Inside: w² / w² = 1 → so we have 8 / (3k⁴)

Square it: 64 / (9k⁸)

Check:
Numerator: (8w²)² = 64w⁴
Denominator: (3w²k⁴)² = 9w⁴k⁸
→ 64w⁴ / 9w⁴k⁸ = 64 / (9k⁸)

✔ Final for #6: 64 / (9k⁸)

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Problem 7:
(6⁴ / 6)³

First, simplify inside: 6⁴ / 6 = 6^(4-1) = 6³

Then cube it: (6³)³ = 6^(9)

✔ Final for #7: 6⁹

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Problem 8:
(3⁴ / 3⁶)²

Inside: 3⁴ / 3⁶ = 3^(4-6) = 3^(-2)

Square it: (3^(-2))² = 3^(-4) = 1 / 3⁴ = 1/81

✔ Final for #8: 1/81

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Problem 9:
(k² / k⁵)³

Inside: k² / k⁵ = k^(-3)

Cube it: (k^(-3))³ = k^(-9) = 1 / k⁹

✔ Final for #9: 1 / k⁹

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Problem 10:
(8r³g² / 2rg⁴)²

Inside:
8/2 = 4
r³ / r = r²
g² / g⁴ = g^(-2)

So inside: 4 × r² × g^(-2) = 4r² / g²

Square it: (4r² / g²)² = 16r⁴ / g⁴

✔ Final for #10: 16r⁴ / g⁴

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Problem 11:
(sb / 7s³b²)²

Inside:
s / s³ = s^(-2)
b / b² = b^(-1)
So: 1/7 × s^(-2) × b^(-1) = 1 / (7s²b)

Square it: 1 / (49s⁴b²)

Check:
Numerator: (sb)² = s²b²
Denominator: (7s³b²)² = 49s⁶b⁴
→ s²b² / 49s⁶b⁴ = 1 / (49s⁴b²)

✔ Final for #11: 1 / (49s⁴b²)

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Problem 12:
(7w² / 8w)²

Inside: w² / w = w¹ → so 7w / 8

Square it: (7w/8)² = 49w² / 64

✔ Final for #12: 49w² / 64

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Final Answer:
1) 125/(512d⁶)
2) 1/c⁹
3) 125g³b¹⁵/64
4) 49/(4r²)
5) 512/(27d⁹g⁶)
6) 64/(9k⁸)
7) 6⁹
8) 1/81
9) 1/k⁹
10) 16r⁴/g⁴
11) 1/(49s⁴b²)
12) 49w²/64