Write the Fraction in Simplest Form ((4v^-3)^3)/(3v^-8)

$\frac{{(4 v^{- 3})}^{3}}{3 v^{- 8}}$

Step 1

Move $v^{- 8}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(4 v^{- 3})}^{3} v^{8}}{3}$

Step 2

Simplify the numerator.

Apply the product rule to $4 v^{- 3}$ .

$\frac{4^{3} {(v^{- 3})}^{3} v^{8}}{3}$

Raise $4$ to the power of $3$ .

$\frac{64 {(v^{- 3})}^{3} v^{8}}{3}$

Multiply the exponents in ${(v^{- 3})}^{3}$ .

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

$\frac{64 v^{- 3 \cdot 3} v^{8}}{3}$

Multiply $- 3$ by $3$ .

$\frac{64 v^{- 9} v^{8}}{3}$

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

$\frac{64 \frac{1}{v^{9}} v^{8}}{3}$

Combine exponents.

Combine $64$ and $\frac{1}{v^{9}}$ .

$\frac{\frac{64}{v^{9}} v^{8}}{3}$

Combine $\frac{64}{v^{9}}$ and $v^{8}$ .

$\frac{\frac{64 v^{8}}{v^{9}}}{3}$

Reduce the expression $\frac{64 v^{8}}{v^{9}}$ by cancelling the common factors.

Factor $v^{8}$ out of $64 v^{8}$ .

$\frac{\frac{v^{8} \cdot 64}{v^{9}}}{3}$

Factor $v^{8}$ out of $v^{9}$ .

$\frac{\frac{v^{8} \cdot 64}{v^{8} v}}{3}$

Cancel the common factor.

$\frac{\frac{v^{8} \cdot 64}{v^{8} v}}{3}$

Rewrite the expression.

$\frac{\frac{64}{v}}{3}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{64}{v} \cdot \frac{1}{3}$

Step 4

Multiply $\frac{64}{v}$ by $\frac{1}{3}$ .

$\frac{64}{v \cdot 3}$

Step 5

Move $3$ to the left of $v$ .

$\frac{64}{3 v}$