Algebra Examples

$\frac{{(m^{- \frac{1}{2}} n^{2})}^{\frac{5}{4}}}{m^{2} n^{\frac{1}{2}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(\frac{1}{m^{\frac{1}{2}}} n^{2})}^{\frac{5}{4}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.2

Combine $\frac{1}{m^{\frac{1}{2}}}$ and $n^{2}$ .

$\frac{{(\frac{n^{2}}{m^{\frac{1}{2}}})}^{\frac{5}{4}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.3

Apply the product rule to $\frac{n^{2}}{m^{\frac{1}{2}}}$ .

$\frac{\frac{{(n^{2})}^{\frac{5}{4}}}{{(m^{\frac{1}{2}})}^{\frac{5}{4}}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.4

Multiply the exponents in ${(n^{2})}^{\frac{5}{4}}$ .

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Step 1.4.1

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

$\frac{\frac{n^{2 (\frac{5}{4})}}{{(m^{\frac{1}{2}})}^{\frac{5}{4}}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Factor $2$ out of $4$ .

$\frac{\frac{n^{2 \frac{5}{2 (2)}}}{{(m^{\frac{1}{2}})}^{\frac{5}{4}}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.4.2.2

Cancel the common factor.

$\frac{\frac{n^{2 \frac{5}{2 \cdot 2}}}{{(m^{\frac{1}{2}})}^{\frac{5}{4}}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.4.2.3

Rewrite the expression.

$\frac{\frac{n^{\frac{5}{2}}}{{(m^{\frac{1}{2}})}^{\frac{5}{4}}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.5

Multiply the exponents in ${(m^{\frac{1}{2}})}^{\frac{5}{4}}$ .

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Step 1.5.1

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

$\frac{\frac{n^{\frac{5}{2}}}{m^{\frac{1}{2} \cdot \frac{5}{4}}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.5.2

Multiply $\frac{1}{2} \cdot \frac{5}{4}$ .

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Step 1.5.2.1

Multiply $\frac{1}{2}$ by $\frac{5}{4}$ .

$\frac{\frac{n^{\frac{5}{2}}}{m^{\frac{5}{2 \cdot 4}}}}{m^{2} n^{\frac{1}{2}}}$

Step 1.5.2.2

Multiply $2$ by $4$ .

$\frac{\frac{n^{\frac{5}{2}}}{m^{\frac{5}{8}}}}{m^{2} n^{\frac{1}{2}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{n^{\frac{5}{2}}}{m^{\frac{5}{8}}} \cdot \frac{1}{m^{2} n^{\frac{1}{2}}}$

Step 3

Combine.

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{\frac{5}{8}} (m^{2} n^{\frac{1}{2}})}$

Step 4

Multiply $m^{\frac{5}{8}}$ by $m^{2}$ by adding the exponents.

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Step 4.1

Move $m^{2}$ .

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{2} m^{\frac{5}{8}} n^{\frac{1}{2}}}$

Step 4.2

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

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{2 + \frac{5}{8}} n^{\frac{1}{2}}}$

Step 4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{2 \cdot \frac{8}{8} + \frac{5}{8}} n^{\frac{1}{2}}}$

Step 4.4

Combine $2$ and $\frac{8}{8}$ .

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{\frac{2 \cdot 8}{8} + \frac{5}{8}} n^{\frac{1}{2}}}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{\frac{2 \cdot 8 + 5}{8}} n^{\frac{1}{2}}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $8$ .

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{\frac{16 + 5}{8}} n^{\frac{1}{2}}}$

Step 4.6.2

Add $16$ and $5$ .

$\frac{n^{\frac{5}{2}} \cdot 1}{m^{\frac{21}{8}} n^{\frac{1}{2}}}$

Step 5

Multiply $n^{\frac{5}{2}}$ by $1$ .

$\frac{n^{\frac{5}{2}}}{m^{\frac{21}{8}} n^{\frac{1}{2}}}$

Step 6

Move $n^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{n^{\frac{5}{2}} n^{- \frac{1}{2}}}{m^{\frac{21}{8}}}$

Step 7

Multiply $n^{\frac{5}{2}}$ by $n^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.1

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

$\frac{n^{\frac{5}{2} - \frac{1}{2}}}{m^{\frac{21}{8}}}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{n^{\frac{5 - 1}{2}}}{m^{\frac{21}{8}}}$

Step 7.3

Subtract $1$ from $5$ .

$\frac{n^{\frac{4}{2}}}{m^{\frac{21}{8}}}$

Step 7.4

Divide $4$ by $2$ .

$\frac{n^{2}}{m^{\frac{21}{8}}}$