Math worksheet for practicing the power rule in exponents.

To solve the given problems using the Power Rule for exponents, we will apply the following rules:

1. Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
2. Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
3. Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)

We will simplify each expression step by step.

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Problem Set:

#### Part 1: Simplify each expression using the Power Rule and write answers in positive exponents.

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#### s) \((m^7 \cdot u^4)^3\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((m^7 \cdot u^4)^3 = (m^7)^3 \cdot (u^4)^3\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((m^7)^3 = m^{7 \cdot 3} = m^{21}\)
- \((u^4)^3 = u^{4 \cdot 3} = u^{12}\)
- Combine the results: \(m^{21} \cdot u^{12}\)

Answer: \(m^{21} \cdot u^{12}\)

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#### t) \((k^6 \cdot g^8)^5\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((k^6 \cdot g^8)^5 = (k^6)^5 \cdot (g^8)^5\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((k^6)^5 = k^{6 \cdot 5} = k^{30}\)
- \((g^8)^5 = g^{8 \cdot 5} = g^{40}\)
- Combine the results: \(k^{30} \cdot g^{40}\)

Answer: \(k^{30} \cdot g^{40}\)

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#### u) \((x^9 \cdot y^7)^2\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((x^9 \cdot y^7)^2 = (x^9)^2 \cdot (y^7)^2\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((x^9)^2 = x^{9 \cdot 2} = x^{18}\)
- \((y^7)^2 = y^{7 \cdot 2} = y^{14}\)
- Combine the results: \(x^{18} \cdot y^{14}\)

Answer: \(x^{18} \cdot y^{14}\)

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#### v) \((h^5 \cdot y^9)^4\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((h^5 \cdot y^9)^4 = (h^5)^4 \cdot (y^9)^4\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((h^5)^4 = h^{5 \cdot 4} = h^{20}\)
- \((y^9)^4 = y^{9 \cdot 4} = y^{36}\)
- Combine the results: \(h^{20} \cdot y^{36}\)

Answer: \(h^{20} \cdot y^{36}\)

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#### w) \((x^8 \cdot d^6)^{-4}\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((x^8 \cdot d^6)^{-4} = (x^8)^{-4} \cdot (d^6)^{-4}\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((x^8)^{-4} = x^{8 \cdot (-4)} = x^{-32}\)
- \((d^6)^{-4} = d^{6 \cdot (-4)} = d^{-24}\)
- Combine the results: \(x^{-32} \cdot d^{-24}\)
- Use the Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
- \(x^{-32} \cdot d^{-24} = \frac{1}{x^{32} \cdot d^{24}}\)

Answer: \(\frac{1}{x^{32} \cdot d^{24}}\)

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#### x) \((2d^3 \cdot t^4)^6\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((2d^3 \cdot t^4)^6 = (2)^6 \cdot (d^3)^6 \cdot (t^4)^6\)
- Simplify each term:
- \((2)^6 = 64\)
- \((d^3)^6 = d^{3 \cdot 6} = d^{18}\)
- \((t^4)^6 = t^{4 \cdot 6} = t^{24}\)
- Combine the results: \(64 \cdot d^{18} \cdot t^{24}\)

Answer: \(64d^{18}t^{24}\)

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#### y) \((u^3 \cdot s^5)^{-2}\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((u^3 \cdot s^5)^{-2} = (u^3)^{-2} \cdot (s^5)^{-2}\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((u^3)^{-2} = u^{3 \cdot (-2)} = u^{-6}\)
- \((s^5)^{-2} = s^{5 \cdot (-2)} = s^{-10}\)
- Combine the results: \(u^{-6} \cdot s^{-10}\)
- Use the Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
- \(u^{-6} \cdot s^{-10} = \frac{1}{u^6 \cdot s^{10}}\)

Answer: \(\frac{1}{u^6 \cdot s^{10}}\)

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#### z) \((h^4 \cdot x^8)^{-3}\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((h^4 \cdot x^8)^{-3} = (h^4)^{-3} \cdot (x^8)^{-3}\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((h^4)^{-3} = h^{4 \cdot (-3)} = h^{-12}\)
- \((x^8)^{-3} = x^{8 \cdot (-3)} = x^{-24}\)
- Combine the results: \(h^{-12} \cdot x^{-24}\)
- Use the Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
- \(h^{-12} \cdot x^{-24} = \frac{1}{h^{12} \cdot x^{24}}\)

Answer: \(\frac{1}{h^{12} \cdot x^{24}}\)

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#### aa) \((p^5 \cdot m^7)^{-4}\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((p^5 \cdot m^7)^{-4} = (p^5)^{-4} \cdot (m^7)^{-4}\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((p^5)^{-4} = p^{5 \cdot (-4)} = p^{-20}\)
- \((m^7)^{-4} = m^{7 \cdot (-4)} = m^{-28}\)
- Combine the results: \(p^{-20} \cdot m^{-28}\)
- Use the Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
- \(p^{-20} \cdot m^{-28} = \frac{1}{p^{20} \cdot m^{28}}\)

Answer: \(\frac{1}{p^{20} \cdot m^{28}}\)

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#### bb) \((2u^3 \cdot z^2)^4\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((2u^3 \cdot z^2)^4 = (2)^4 \cdot (u^3)^4 \cdot (z^2)^4\)
- Simplify each term:
- \((2)^4 = 16\)
- \((u^3)^4 = u^{3 \cdot 4} = u^{12}\)
- \((z^2)^4 = z^{2 \cdot 4} = z^8\)
- Combine the results: \(16 \cdot u^{12} \cdot z^8\)

Answer: \(16u^{12}z^8\)

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#### cc) \((b^4 \cdot u^3)^2\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((b^4 \cdot u^3)^2 = (b^4)^2 \cdot (u^3)^2\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((b^4)^2 = b^{4 \cdot 2} = b^8\)
- \((u^3)^2 = u^{3 \cdot 2} = u^6\)
- Combine the results: \(b^8 \cdot u^6\)

Answer: \(b^8 \cdot u^6\)

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#### dd) \((m^3 \cdot c^8)^{-4}\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((m^3 \cdot c^8)^{-4} = (m^3)^{-4} \cdot (c^8)^{-4}\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((m^3)^{-4} = m^{3 \cdot (-4)} = m^{-12}\)
- \((c^8)^{-4} = c^{8 \cdot (-4)} = c^{-32}\)
- Combine the results: \(m^{-12} \cdot c^{-32}\)
- Use the Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
- \(m^{-12} \cdot c^{-32} = \frac{1}{m^{12} \cdot c^{32}}\)

Answer: \(\frac{1}{m^{12} \cdot c^{32}}\)

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#### ee) \((2g^3)^4\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((2g^3)^4 = (2)^4 \cdot (g^3)^4\)
- Simplify each term:
- \((2)^4 = 16\)
- \((g^3)^4 = g^{3 \cdot 4} = g^{12}\)
- Combine the results: \(16 \cdot g^{12}\)

Answer: \(16g^{12}\)

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#### ff) \((q^3 \cdot n^4)^{-2}\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((q^3 \cdot n^4)^{-2} = (q^3)^{-2} \cdot (n^4)^{-2}\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((q^3)^{-2} = q^{3 \cdot (-2)} = q^{-6}\)
- \((n^4)^{-2} = n^{4 \cdot (-2)} = n^{-8}\)
- Combine the results: \(q^{-6} \cdot n^{-8}\)
- Use the Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
- \(q^{-6} \cdot n^{-8} = \frac{1}{q^6 \cdot n^8}\)

Answer: \(\frac{1}{q^6 \cdot n^8}\)

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#### gg) \((c^3 \cdot t^4)^{-1}\)
- Apply the Power of a Product Rule: \((ab)^n = a^n \cdot b^n\)
- \((c^3 \cdot t^4)^{-1} = (c^3)^{-1} \cdot (t^4)^{-1}\)
- Apply the Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
- \((c^3)^{-1} = c^{3 \cdot (-1)} = c^{-3}\)
- \((t^4)^{-1} = t^{4 \cdot (-1)} = t^{-4}\)
- Combine the results: \(c^{-3} \cdot t^{-4}\)
- Use the Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)
- \(c^{-3} \cdot t^{-4} = \frac{1}{c^3 \cdot t^4}\)

Answer: \(\frac{1}{c^3 \cdot t^4}\)

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Final Answers:

\[
\boxed{
\begin{aligned}
&\text{s) } m^{21} \cdot u^{12} \\
&\text{t) } k^{30} \cdot g^{40} \\
&\text{u) } x^{18} \cdot y^{14} \\
&\text{v) } h^{20} \cdot y^{36} \\
&\text{w) } \frac{1}{x^{32} \cdot d^{24}} \\
&\text{x) } 64d^{18}t^{24} \\
&\text{y) } \frac{1}{u^6 \cdot s^{10}} \\
&\text{z) } \frac{1}{h^{12} \cdot x^{24}} \\
&\text{aa) } \frac{1}{p^{20} \cdot m^{28}} \\
&\text{bb) } 16u^{12}z^8 \\
&\text{cc) } b^8 \cdot u^6 \\
&\text{dd) } \frac{1}{m^{12} \cdot c^{32}} \\
&\text{ee) } 16g^{12} \\
&\text{ff) } \frac{1}{q^6 \cdot n^8} \\
&\text{gg) } \frac{1}{c^3 \cdot t^4}
\end{aligned}
}
\]