Finite Math Examples

${(\frac{r^{\frac{5}{6}}}{r^{\frac{2}{3}}})}^{\frac{4}{7}}$

Step 1

Move $r^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(r^{\frac{5}{6}} r^{- \frac{2}{3}})}^{\frac{4}{7}}$

Step 2

Multiply $r^{\frac{5}{6}}$ by $r^{- \frac{2}{3}}$ by adding the exponents.

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Step 2.1

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

${(r^{\frac{5}{6} - \frac{2}{3}})}^{\frac{4}{7}}$

Step 2.2

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(r^{\frac{5}{6} - \frac{2}{3} \cdot \frac{2}{2}})}^{\frac{4}{7}}$

Step 2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

${(r^{\frac{5}{6} - \frac{2 \cdot 2}{3 \cdot 2}})}^{\frac{4}{7}}$

Step 2.3.2

Multiply $3$ by $2$ .

${(r^{\frac{5}{6} - \frac{2 \cdot 2}{6}})}^{\frac{4}{7}}$

Step 2.4

Combine the numerators over the common denominator.

${(r^{\frac{5 - 2 \cdot 2}{6}})}^{\frac{4}{7}}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 2$ by $2$ .

${(r^{\frac{5 - 4}{6}})}^{\frac{4}{7}}$

Step 2.5.2

Subtract $4$ from $5$ .

${(r^{\frac{1}{6}})}^{\frac{4}{7}}$

Step 3

Multiply the exponents in ${(r^{\frac{1}{6}})}^{\frac{4}{7}}$ .

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Step 3.1

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

$r^{\frac{1}{6} \cdot \frac{4}{7}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Factor $2$ out of $6$ .

$r^{\frac{1}{2 (3)} \cdot \frac{4}{7}}$

Step 3.2.2

Factor $2$ out of $4$ .

$r^{\frac{1}{2 \cdot 3} \cdot \frac{2 \cdot 2}{7}}$

Step 3.2.3

Cancel the common factor.

$r^{\frac{1}{2 \cdot 3} \cdot \frac{2 \cdot 2}{7}}$

Step 3.2.4

Rewrite the expression.

$r^{\frac{1}{3} \cdot \frac{2}{7}}$

Step 3.3

Multiply $\frac{1}{3}$ by $\frac{2}{7}$ .

$r^{\frac{2}{3 \cdot 7}}$

Step 3.4

Multiply $3$ by $7$ .

$r^{\frac{2}{21}}$