Algebra Examples

${(\frac{2 p}{q^{2}})}^{5} {(\frac{3 p^{7}}{q^{- 7}})}^{- 1}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $\frac{2 p}{q^{2}}$ .

$\frac{{(2 p)}^{5}}{{(q^{2})}^{5}} {(\frac{3 p^{7}}{q^{- 7}})}^{- 1}$

Step 1.2

Apply the product rule to $2 p$ .

$\frac{2^{5} p^{5}}{{(q^{2})}^{5}} {(\frac{3 p^{7}}{q^{- 7}})}^{- 1}$

Step 2

Raise $2$ to the power of $5$ .

$\frac{32 p^{5}}{{(q^{2})}^{5}} {(\frac{3 p^{7}}{q^{- 7}})}^{- 1}$

Step 3

Multiply the exponents in ${(q^{2})}^{5}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{32 p^{5}}{q^{2 \cdot 5}} {(\frac{3 p^{7}}{q^{- 7}})}^{- 1}$

Step 3.2

Multiply $2$ by $5$ .

$\frac{32 p^{5}}{q^{10}} {(\frac{3 p^{7}}{q^{- 7}})}^{- 1}$

Step 4

Move $q^{- 7}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{32 p^{5}}{q^{10}} {(3 p^{7} q^{7})}^{- 1}$

Step 5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{32 p^{5}}{q^{10}} \cdot \frac{1}{3 p^{7} q^{7}}$

Step 6

Combine.

$\frac{32 p^{5} \cdot 1}{q^{10} (3 p^{7} q^{7})}$

Step 7

Multiply $q^{10}$ by $q^{7}$ by adding the exponents.

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Step 7.1

Move $q^{7}$ .

$\frac{32 p^{5} \cdot 1}{q^{7} q^{10} (3 p^{7})}$

Step 7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{32 p^{5} \cdot 1}{q^{7 + 10} (3 p^{7})}$

Step 7.3

Add $7$ and $10$ .

$\frac{32 p^{5} \cdot 1}{q^{17} (3 p^{7})}$

Step 8

Multiply $32 p^{5}$ by $1$ .

$\frac{32 p^{5}}{q^{17} (3 p^{7})}$

Step 9

Cancel the common factor of $p^{5}$ and $p^{7}$ .

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Step 9.1

Factor $p^{5}$ out of $32 p^{5}$ .

$\frac{p^{5} \cdot 32}{q^{17} (3 p^{7})}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $p^{5}$ out of $q^{17} (3 p^{7})$ .

$\frac{p^{5} \cdot 32}{p^{5} (q^{17} (3 p^{2}))}$

Step 9.2.2

Cancel the common factor.

$\frac{p^{5} \cdot 32}{p^{5} (q^{17} (3 p^{2}))}$

Step 9.2.3

Rewrite the expression.

$\frac{32}{q^{17} (3 p^{2})}$

Step 10

Move $3$ to the left of $q^{17}$ .

$\frac{32}{3 q^{17} p^{2}}$