Simplify (m^(-3/4)n^(-1/4))/((m^2n)^(1/2))

$\frac{m^{- \frac{3}{4}} n^{- \frac{1}{4}}}{{(m^{2} n)}^{\frac{1}{2}}}$

Step 1

Simplify the expression.

Move $m^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{n^{- \frac{1}{4}}}{{(m^{2} n)}^{\frac{1}{2}} m^{\frac{3}{4}}}$

Move $n^{- \frac{1}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(m^{2} n)}^{\frac{1}{2}} m^{\frac{3}{4}} n^{\frac{1}{4}}}$

Step 2

Simplify the denominator.

Apply the product rule to $m^{2} n$ .

$\frac{1}{{(m^{2})}^{\frac{1}{2}} n^{\frac{1}{2}} m^{\frac{3}{4}} n^{\frac{1}{4}}}$

Multiply the exponents in ${(m^{2})}^{\frac{1}{2}}$ .

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

$\frac{1}{m^{2 (\frac{1}{2})} n^{\frac{1}{2}} m^{\frac{3}{4}} n^{\frac{1}{4}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{m^{2 (\frac{1}{2})} n^{\frac{1}{2}} m^{\frac{3}{4}} n^{\frac{1}{4}}}$

Rewrite the expression.

$\frac{1}{m^{1} n^{\frac{1}{2}} m^{\frac{3}{4}} n^{\frac{1}{4}}}$

Simplify.

$\frac{1}{m n^{\frac{1}{2}} m^{\frac{3}{4}} n^{\frac{1}{4}}}$

Combine exponents.

Multiply $m$ by $m^{\frac{3}{4}}$ by adding the exponents.

Move $m^{\frac{3}{4}}$ .

$\frac{1}{m^{\frac{3}{4}} m n^{\frac{1}{2}} n^{\frac{1}{4}}}$

Multiply $m^{\frac{3}{4}}$ by $m$ .

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

$\frac{1}{m^{\frac{3}{4}} m^{1} n^{\frac{1}{2}} n^{\frac{1}{4}}}$

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

$\frac{1}{m^{\frac{3}{4} + 1} n^{\frac{1}{2}} n^{\frac{1}{4}}}$

Write $1$ as a fraction with a common denominator.

$\frac{1}{m^{\frac{3}{4} + \frac{4}{4}} n^{\frac{1}{2}} n^{\frac{1}{4}}}$

Combine the numerators over the common denominator.

$\frac{1}{m^{\frac{3 + 4}{4}} n^{\frac{1}{2}} n^{\frac{1}{4}}}$

Add $3$ and $4$ .

$\frac{1}{m^{\frac{7}{4}} n^{\frac{1}{2}} n^{\frac{1}{4}}}$

Multiply $n^{\frac{1}{2}}$ by $n^{\frac{1}{4}}$ by adding the exponents.

Move $n^{\frac{1}{4}}$ .

$\frac{1}{m^{\frac{7}{4}} (n^{\frac{1}{4}} n^{\frac{1}{2}})}$

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

$\frac{1}{m^{\frac{7}{4}} n^{\frac{1}{4} + \frac{1}{2}}}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{m^{\frac{7}{4}} n^{\frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}}}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{1}{m^{\frac{7}{4}} n^{\frac{1}{4} + \frac{2}{2 \cdot 2}}}$

Multiply $2$ by $2$ .

$\frac{1}{m^{\frac{7}{4}} n^{\frac{1}{4} + \frac{2}{4}}}$

Combine the numerators over the common denominator.

$\frac{1}{m^{\frac{7}{4}} n^{\frac{1 + 2}{4}}}$

Add $1$ and $2$ .

$\frac{1}{m^{\frac{7}{4}} n^{\frac{3}{4}}}$

Simplify (m^(-3/4)n^(-1/4))/((m^2n)^(1/2))

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