## How to Solve Powers of Products and Quotients

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### Discovering a Product’s Power

In order to simplify a product’s power in $$2$$ exponential expressions, one can utilize the power of a product rule of exponents. This separates the power of a product of factors into the product of the powers of the factors. For example, look at $$(pq)^3$$. You start via utilizing the associative and commutative properties of multiplication for regrouping the factors.

$$(pq)^3 \ = \ (pq) \ \times \ (pq) \ \times \ (pq) \ = \ p \ \times \ p \ \times \ p \ \times \ q \ \times \ q \ \times \ q \ = \ p^3 \ \times \ q^3$$

So, $$(pq)^3 \ = \ p^3 \ \times \ q^3$$

### THE POWER OF A PRODUCT RULE OF EXPONENTS

With any actual numbers $$a$$ as well as $$b$$ and any integer $$n$$, the power of a product rule of exponents says:

$$(ab)^n \ = \ a^n \ \times \ b^n$$

### Discovering a Quotient’s Power

In order to simplify the power of a quotient with $$2$$ expressions, one can utilize the power of a quotient rule. It says the power of a quotient of factors equals the quotient of the powers of the factors. For instance, here is an example:
$$(e^{-2} \ \times \ f^2)^7 \ = \ \frac{f^{14}}{e^{14}}$$
Now redo the original problem in a different way and see the outcome.
$$(e^{-2} \ \times \ f^2)^7 \ = \ (\frac{f^2}{e^2} \ )^7 \ = \ \frac{f^{14}}{e^{14}}$$
Based on the last $$2$$ steps, it seems one can utilize the power of a product rule for a power of a quotient rule.
$$(e^{-2} \ \times \ f^2)^7 \ = \ (\frac{f^2}{e^2} \ )^7 \ = \ \frac{(f^2)^7}{(e^2)^7} \ = \ \frac{f^{2 \times 7}}{e^{2 \times 7}} \ = \ \frac{f^{14}}{e^{14}}$$

### THE POWER OF A QUOTIENT RULE OF EXPONENTS

With any true numbers $$a$$ and $$b$$ and any integer $$n$$, the power of a quotient rule of exponents says that: $$(\frac{a}{b} \ )^n \ = \ \frac{a^n}{b^n}$$

### Exercises for Powers of Products and Quotients

1) $$(-3x^2x^2)^{3} =$$

2) $$(4x^2x^2)^{3} =$$

3) $$(-9x^3y)^{2} =$$

4) $$(6x^2)^{3} =$$

5) $$(-9x^2)^{3} =$$

6) $$(-2x^4y)^{2} =$$

7) $$(-x^3y)^{2} =$$

8) $$(-8x^2)^{3} =$$

9) $$(6x^4y)^{3} =$$

10) $$(-x^3)^{2} =$$

1) $$(-3x^2x^2)^{3} =$$$$\ \color{red}{(-3^3)x^{(2 \times 3)}x^{(2 \times 3)}}$$$$\ \color{red}{= -27x^{12}}$$
2) $$(4x^2x^2)^{3} =$$$$\ \color{red}{(4^3)x^{(2 \times 3)}x^{(2 \times 3)}}$$$$\ \color{red}{= 64x^{12}}$$
3) $$(-9x^3y)^{2} =$$$$\ \color{red}{(-9^2)x^{(3 \times 2)}y^{2}}$$$$\ \color{red}{= 81x^{6}y^{2}}$$
4) $$(6x^2)^{3} =$$$$\ \color{red}{(6^3)x^{(2 \times 3)}}$$$$\ \color{red}{= 216x^{6}}$$
5) $$(-9x^2)^{3} =$$$$\ \color{red}{(-9^3)x^{(2 \times 3)}}$$$$\ \color{red}{= -729x^{6}}$$
6) $$(-2x^4y)^{2} =$$$$\ \color{red}{(-2^2)x^{(4 \times 2)}y^{2}}$$$$\ \color{red}{= 4x^{8}y^{2}}$$
7) $$(-x^3y)^{2} =$$$$\ \color{red}{(-1^2)x^{(3 \times 2)}y^{2}}$$$$\ \color{red}{= x^{6}y^{2}}$$
8) $$(-8x^2)^{3} =$$$$\ \color{red}{(-8^3)x^{(2 \times 3)}}$$$$\ \color{red}{= -512x^{6}}$$
9) $$(6x^4y)^{3} =$$$$\ \color{red}{(6^3)x^{(4 \times 3)}y^{3}}$$$$\ \color{red}{= 216x^{12}y^{3}}$$
10) $$(-x^3)^{2} =$$$$\ \color{red}{(-1^2)x^{(3 \times 2)}}$$$$\ \color{red}{= x^{6}}$$

## Powers of Products and Quotients Quiz

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