Exponent Rules - Math Worksheets - Free Printable

The image you've uploaded is a math worksheet titled "Exponent Rules: Quotient To A Power Rule", and it includes an answer key. The goal of the worksheet is to simplify expressions using the quotient to a power rule of exponents.

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🔷 Quotient to a Power Rule:

This rule states that:

$$
\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}
$$

In other words, when you raise a quotient (fraction) to a power, you apply the exponent to both the numerator and the denominator separately.

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Let’s go through each problem step-by-step to verify the answers provided in the key.

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✔ Problem 1: $\left(\frac{x^2}{y^3}\right)^4$

Solution:
$$
\left(\frac{x^2}{y^3}\right)^4 = \frac{(x^2)^4}{(y^3)^4} = \frac{x^{8}}{y^{12}}
$$
✔ Answer: $\boxed{\frac{x^8}{y^{12}}}$ — Correct.

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✔ Problem 2: $\left(\frac{a^3}{b^2}\right)^5$

$$
= \frac{(a^3)^5}{(b^2)^5} = \frac{a^{15}}{b^{10}}
$$
✔ Answer: $\boxed{\frac{a^{15}}{b^{10}}}$ — Correct.

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✔ Problem 3: $\left(\frac{m^4}{n^3}\right)^2$

$$
= \frac{(m^4)^2}{(n^3)^2} = \frac{m^8}{n^6}
$$
✔ Answer: $\boxed{\frac{m^8}{n^6}}$ — Correct.

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✔ Problem 4: $\left(\frac{c^5}{d^4}\right)^3$

$$
= \frac{(c^5)^3}{(d^4)^3} = \frac{c^{15}}{d^{12}}
$$
✔ Answer: $\boxed{\frac{c^{15}}{d^{12}}}$ — Correct.

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✔ Problem 5: $\left(\frac{y^2}{z^3}\right)^4$

$$
= \frac{(y^2)^4}{(z^3)^4} = \frac{y^8}{z^{12}}
$$
✔ Answer: $\boxed{\frac{y^8}{z^{12}}}$ — Correct.

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✔ Problem 6: $\left(\frac{g^7}{r^5}\right)^3$

$$
= \frac{(g^7)^3}{(r^5)^3} = \frac{g^{21}}{r^{15}}
$$
✔ Answer: $\boxed{\frac{g^{21}}{r^{15}}}$ — Correct.

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✔ Problem 7: $\left(\frac{p^3}{q^4}\right)^5$

$$
= \frac{(p^3)^5}{(q^4)^5} = \frac{p^{15}}{q^{20}}
$$
✔ Answer: $\boxed{\frac{p^{15}}{q^{20}}}$ — Correct.

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✔ Problem 8: $\left(\frac{g^5}{h^7}\right)^3$

$$
= \frac{(g^5)^3}{(h^7)^3} = \frac{g^{15}}{h^{21}}
$$
✔ Answer: $\boxed{\frac{g^{15}}{h^{21}}}$ — Correct.

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✔ Problem 9: $\left(\frac{t^6}{s^3}\right)^4$

$$
= \frac{(t^6)^4}{(s^3)^4} = \frac{t^{24}}{s^{12}}
$$
✔ Answer: $\boxed{\frac{t^{24}}{s^{12}}}$ — Correct.

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✔ Problem 10: $\left(\frac{b^3}{c^2}\right)^6$

$$
= \frac{(b^3)^6}{(c^2)^6} = \frac{b^{18}}{c^{12}}
$$
✔ Answer: $\boxed{\frac{b^{18}}{c^{12}}}$ — Correct.

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✔ Problem 11: $\left(\frac{x^4}{y^3}\right)^5$

$$
= \frac{(x^4)^5}{(y^3)^5} = \frac{x^{20}}{y^{15}}
$$
✔ Answer: $\boxed{\frac{x^{20}}{y^{15}}}$ — Correct.

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✔ Problem 12: $\left(\frac{i^5}{k^3}\right)^2$

$$
= \frac{(i^5)^2}{(k^3)^2} = \frac{i^{10}}{k^6}
$$
✔ Answer: $\boxed{\frac{i^{10}}{k^6}}$ — Correct.

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✔ Final Summary:

All the answers on the answer key are correct and follow the quotient to a power rule properly.

Each expression was simplified by:
- Applying the outer exponent to both the numerator and denominator.
- Using the power of a power rule: $(a^m)^n = a^{m \cdot n}$.

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📌 Key Takeaway:

When simplifying $\left(\frac{a^m}{b^n}\right)^p$, the result is:
$$
\frac{a^{m \cdot p}}{b^{n \cdot p}}
$$

This rule is essential for working with algebraic expressions involving exponents.

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Let me know if you'd like a printable version or explanation tailored for students!