Simplify powers of products and quotients with this algebra worksheet.

Sure! Let’s solve each problem step by step using the Power of a Product Rule and Power of a Power Rule.

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📌 Key Rules to Remember:

1. Power of a Product:
\((ab)^n = a^n \cdot b^n\)

2. Power of a Power:
\((a^m)^n = a^{m \cdot n}\)

3. Negative Exponent:
\(a^{-n} = \frac{1}{a^n}\)

4. Multiply like bases:
\(a^m \cdot a^n = a^{m+n}\) — used in problems where you combine terms inside parentheses first.

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Let’s go one by one:

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1) \((3a^2)^3\)

→ Apply power to both 3 and \(a^2\):
= \(3^3 \cdot (a^2)^3 = 27 \cdot a^{6} = \boxed{27a^6}\)

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2) \((2n^4)^4\)

= \(2^4 \cdot (n^4)^4 = 16 \cdot n^{16} = \boxed{16n^{16}}\)

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3) \((3^3 \times 2^2)^3\)

First, compute inside: \(3^3 = 27\), \(2^2 = 4\) → \(27 \times 4 = 108\)
Then: \(108^3\) — but better to keep in exponential form:

= \((3^3 \cdot 2^2)^3 = 3^{9} \cdot 2^{6} = \boxed{3^9 \cdot 2^6}\)
*(You can leave it like this, or compute numerically: 19683 × 64 = 1,259,712 — but exponential form is preferred unless asked for number.)*

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4) \((3x^4)^4\)

= \(3^4 \cdot (x^4)^4 = 81 \cdot x^{16} = \boxed{81x^{16}}\)

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5) \((6b^2)^2\)

= \(6^2 \cdot (b^2)^2 = 36 \cdot b^4 = \boxed{36b^4}\)

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6) \((11x^5)^2\)

= \(11^2 \cdot (x^5)^2 = 121 \cdot x^{10} = \boxed{121x^{10}}\)

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7) \((7y^4)^2\)

= \(7^2 \cdot (y^4)^2 = 49 \cdot y^8 = \boxed{49y^8}\)

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8) \((3ab^4)^4\)

= \(3^4 \cdot a^4 \cdot (b^4)^4 = 81 \cdot a^4 \cdot b^{16} = \boxed{81a^4b^{16}}\)

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9) \((2^5)^2\)

= \(2^{5 \cdot 2} = 2^{10} = \boxed{1024}\)
*(or leave as \(2^{10}\))*

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10) \((2x^4y^4)^3\)

= \(2^3 \cdot (x^4)^3 \cdot (y^4)^3 = 8 \cdot x^{12} \cdot y^{12} = \boxed{8x^{12}y^{12}}\)

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11) \((5m)^3\)

= \(5^3 \cdot m^3 = 125m^3 = \boxed{125m^3}\)

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12) \((2m)^3\)

= \(2^3 \cdot m^3 = 8m^3 = \boxed{8m^3}\)

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13) \((x^2y^2)^2\)

= \((x^2)^2 \cdot (y^2)^2 = x^4 \cdot y^4 = \boxed{x^4y^4}\)

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14) \((6x^4)^2\)

= \(6^2 \cdot (x^4)^2 = 36x^8 = \boxed{36x^8}\)

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15) \((6x^{10})^2\)

= \(6^2 \cdot (x^{10})^2 = 36x^{20} = \boxed{36x^{20}}\)

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16) \((u^4v^3)^2\)

= \((u^4)^2 \cdot (v^3)^2 = u^8 \cdot v^6 = \boxed{u^8v^6}\)

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17) \((2x^4)^5\)

= \(2^5 \cdot (x^4)^5 = 32 \cdot x^{20} = \boxed{32x^{20}}\)

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18) \((2x^5)^3\)

= \(2^3 \cdot (x^5)^3 = 8 \cdot x^{15} = \boxed{8x^{15}}\)

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19) \((3x^2 \cdot 2x^2)^2\)

First, simplify inside:
\(3x^2 \cdot 2x^2 = (3 \cdot 2) \cdot (x^2 \cdot x^2) = 6x^4\)
Now raise to power 2:
\((6x^4)^2 = 6^2 \cdot (x^4)^2 = 36x^8 = \boxed{36x^8}\)

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20) \((2p^3 \cdot 2p)^2\)

Simplify inside:
\(2p^3 \cdot 2p = (2 \cdot 2) \cdot (p^3 \cdot p) = 4p^4\)
Now square:
\((4p^4)^2 = 4^2 \cdot (p^4)^2 = 16p^8 = \boxed{16p^8}\)

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21) \((2n^3)^{-2}\)

Apply negative exponent rule:
= \(\frac{1}{(2n^3)^2} = \frac{1}{2^2 \cdot (n^3)^2} = \frac{1}{4n^6} = \boxed{\frac{1}{4n^6}}\)

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## ✔ Final Answers Summary:

1) \(27a^6\)
2) \(16n^{16}\)
3) \(3^9 \cdot 2^6\) (or 1,259,712)
4) \(81x^{16}\)
5) \(36b^4\)
6) \(121x^{10}\)
7) \(49y^8\)
8) \(81a^4b^{16}\)
9) \(2^{10}\) or 1024
10) \(8x^{12}y^{12}\)
11) \(125m^3\)
12) \(8m^3\)
13) \(x^4y^4\)
14) \(36x^8\)
15) \(36x^{20}\)
16) \(u^8v^6\)
17) \(32x^{20}\)
18) \(8x^{15}\)
19) \(36x^8\)
20) \(16p^8\)
21) \(\frac{1}{4n^6}\)

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Let me know if you’d like these written out with more detailed steps or formatted for printing! 😊