Algebra Examples

$\frac{m^{\frac{1}{5}} n^{\frac{1}{7}}}{m^{\frac{1}{4}} n^{\frac{2}{7}}}$

Step 1

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

$\frac{n^{\frac{1}{7}}}{m^{\frac{1}{4}} n^{\frac{2}{7}} m^{- \frac{1}{5}}}$

Step 2

Move $n^{\frac{1}{7}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{m^{\frac{1}{4}} n^{\frac{2}{7}} m^{- \frac{1}{5}} n^{- \frac{1}{7}}}$

Step 3

Simplify the denominator.

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Step 3.1

Multiply $m^{\frac{1}{4}}$ by $m^{- \frac{1}{5}}$ by adding the exponents.

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Step 3.1.1

Move $m^{- \frac{1}{5}}$ .

$\frac{1}{m^{- \frac{1}{5}} m^{\frac{1}{4}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.2

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

$\frac{1}{m^{- \frac{1}{5} + \frac{1}{4}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.3

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{m^{- \frac{1}{5} \cdot \frac{4}{4} + \frac{1}{4}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.4

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{m^{- \frac{1}{5} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{5}{5}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.5

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.5.1

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

$\frac{1}{m^{- \frac{4}{5 \cdot 4} + \frac{1}{4} \cdot \frac{5}{5}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.5.2

Multiply $5$ by $4$ .

$\frac{1}{m^{- \frac{4}{20} + \frac{1}{4} \cdot \frac{5}{5}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.5.3

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

$\frac{1}{m^{- \frac{4}{20} + \frac{5}{4 \cdot 5}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.5.4

Multiply $4$ by $5$ .

$\frac{1}{m^{- \frac{4}{20} + \frac{5}{20}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.6

Combine the numerators over the common denominator.

$\frac{1}{m^{\frac{- 4 + 5}{20}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.1.7

Add $- 4$ and $5$ .

$\frac{1}{m^{\frac{1}{20}} n^{\frac{2}{7}} n^{- \frac{1}{7}}}$

Step 3.2

Multiply $n^{\frac{2}{7}}$ by $n^{- \frac{1}{7}}$ by adding the exponents.

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Step 3.2.1

Move $n^{- \frac{1}{7}}$ .

$\frac{1}{m^{\frac{1}{20}} (n^{- \frac{1}{7}} n^{\frac{2}{7}})}$

Step 3.2.2

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

$\frac{1}{m^{\frac{1}{20}} n^{- \frac{1}{7} + \frac{2}{7}}}$

Step 3.2.3

Combine the numerators over the common denominator.

$\frac{1}{m^{\frac{1}{20}} n^{\frac{- 1 + 2}{7}}}$

Step 3.2.4

Add $- 1$ and $2$ .

$\frac{1}{m^{\frac{1}{20}} n^{\frac{1}{7}}}$