Simplify (uv)/((2u^4v^-2)^2*(u^2v^-1))

$\frac{u v}{{(2 u^{4} v^{- 2})}^{2} \cdot (u^{2} v^{- 1})}$

Step 1

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

$\frac{u v \cdot v}{{(2 u^{4} v^{- 2})}^{2} \cdot u^{2}}$

Step 2

Multiply $v$ by $v$ by adding the exponents.

Move $v$ .

$\frac{u (v \cdot v)}{{(2 u^{4} v^{- 2})}^{2} \cdot u^{2}}$

Multiply $v$ by $v$ .

$\frac{u v^{2}}{{(2 u^{4} v^{- 2})}^{2} \cdot u^{2}}$

Step 3

Cancel the common factor of $u$ and $u^{2}$ .

Factor $u$ out of $u v^{2}$ .

$\frac{u (v^{2})}{{(2 u^{4} v^{- 2})}^{2} \cdot u^{2}}$

Cancel the common factors.

Factor $u$ out of ${(2 u^{4} v^{- 2})}^{2} \cdot u^{2}$ .

$\frac{u (v^{2})}{u ({(2 u^{4} v^{- 2})}^{2} \cdot u)}$

Cancel the common factor.

$\frac{u v^{2}}{u ({(2 u^{4} v^{- 2})}^{2} \cdot u)}$

Rewrite the expression.

$\frac{v^{2}}{{(2 u^{4} v^{- 2})}^{2} \cdot u}$

Step 4

Simplify the denominator.

Apply the product rule to $2 u^{4} v^{- 2}$ .

$\frac{v^{2}}{{(2 u^{4})}^{2} {(v^{- 2})}^{2} u}$

Apply the product rule to $2 u^{4}$ .

$\frac{v^{2}}{2^{2} {(u^{4})}^{2} {(v^{- 2})}^{2} u}$

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

$\frac{v^{2}}{4 {(u^{4})}^{2} {(v^{- 2})}^{2} u}$

Multiply the exponents in ${(u^{4})}^{2}$ .

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

$\frac{v^{2}}{4 u^{4 \cdot 2} {(v^{- 2})}^{2} u}$

Multiply $4$ by $2$ .

$\frac{v^{2}}{4 u^{8} {(v^{- 2})}^{2} u}$

Multiply the exponents in ${(v^{- 2})}^{2}$ .

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

$\frac{v^{2}}{4 u^{8} v^{- 2 \cdot 2} u}$

Multiply $- 2$ by $2$ .

$\frac{v^{2}}{4 u^{8} v^{- 4} u}$

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

$\frac{v^{2}}{4 u^{8} \frac{1}{v^{4}} u}$

Combine exponents.

Combine $4$ and $\frac{1}{v^{4}}$ .

$\frac{v^{2}}{u^{8} \frac{4}{v^{4}} u}$

Combine $u^{8}$ and $\frac{4}{v^{4}}$ .

$\frac{v^{2}}{\frac{u^{8} \cdot 4}{v^{4}} u}$

Combine $\frac{u^{8} \cdot 4}{v^{4}}$ and $u$ .

$\frac{v^{2}}{\frac{u^{8} \cdot 4 u}{v^{4}}}$

Raise $u$ to the power of $1$ .

$\frac{v^{2}}{\frac{4 (u^{8} u^{1})}{v^{4}}}$

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

$\frac{v^{2}}{\frac{4 u^{8 + 1}}{v^{4}}}$

Add $8$ and $1$ .

$\frac{v^{2}}{\frac{4 u^{9}}{v^{4}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$v^{2} \frac{v^{4}}{4 u^{9}}$

Step 6

Multiply $v^{2} \frac{v^{4}}{4 u^{9}}$ .

Combine $v^{2}$ and $\frac{v^{4}}{4 u^{9}}$ .

$\frac{v^{2} v^{4}}{4 u^{9}}$

Multiply $v^{2}$ by $v^{4}$ by adding the exponents.

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

$\frac{v^{2 + 4}}{4 u^{9}}$

Add $2$ and $4$ .

$\frac{v^{6}}{4 u^{9}}$

Simplify (uv)/((2u^4v^-2)^2*(u^2v^-1))

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