Simplify ((2pi)/(q^2))^3((3pi^5)/(q^-5))^-1

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

Step 1

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

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

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

Apply the product rule to $2 π$ .

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

Step 2

Raise $2$ to the power of $3$ .

$\frac{8 π^{3}}{{(q^{2})}^{3}} {(\frac{3 π^{5}}{q^{- 5}})}^{- 1}$

Step 3

Multiply the exponents in ${(q^{2})}^{3}$ .

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

$\frac{8 π^{3}}{q^{2 \cdot 3}} {(\frac{3 π^{5}}{q^{- 5}})}^{- 1}$

Multiply $2$ by $3$ .

$\frac{8 π^{3}}{q^{6}} {(\frac{3 π^{5}}{q^{- 5}})}^{- 1}$

Step 4

Move $q^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{8 π^{3}}{q^{6}} {(3 π^{5} q^{5})}^{- 1}$

Step 5

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

$\frac{8 π^{3}}{q^{6}} \cdot \frac{1}{3 π^{5} q^{5}}$

Step 6

Combine.

$\frac{8 π^{3} \cdot 1}{q^{6} (3 π^{5} q^{5})}$

Step 7

Multiply $q^{6}$ by $q^{5}$ by adding the exponents.

Move $q^{5}$ .

$\frac{8 π^{3} \cdot 1}{q^{5} q^{6} (3 π^{5})}$

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

$\frac{8 π^{3} \cdot 1}{q^{5 + 6} (3 π^{5})}$

Add $5$ and $6$ .

$\frac{8 π^{3} \cdot 1}{q^{11} (3 π^{5})}$

Step 8

Multiply $8 π^{3}$ by $1$ .

$\frac{8 π^{3}}{q^{11} (3 π^{5})}$

Step 9

Cancel the common factor of $π^{3}$ and $π^{5}$ .

Factor $π^{3}$ out of $8 π^{3}$ .

$\frac{π^{3} \cdot 8}{q^{11} (3 π^{5})}$

Cancel the common factors.

Factor $π^{3}$ out of $q^{11} (3 π^{5})$ .

$\frac{π^{3} \cdot 8}{π^{3} (q^{11} (3 π^{2}))}$

Cancel the common factor.

$\frac{π^{3} \cdot 8}{π^{3} (q^{11} (3 π^{2}))}$

Rewrite the expression.

$\frac{8}{q^{11} (3 π^{2})}$

Step 10

Move $3$ to the left of $q^{11}$ .

$\frac{8}{3 q^{11} π^{2}}$

Simplify ((2pi)/(q^2))^3((3pi^5)/(q^-5))^-1

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