Algebra Examples

${(\frac{16^{\frac{5}{3}}}{16^{\frac{5}{6}}})}^{\frac{9}{5}}$

Step 1

Move $16^{\frac{5}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(16^{\frac{5}{3}} \cdot 16^{- \frac{5}{6}})}^{\frac{9}{5}}$

Step 2

Multiply $16^{\frac{5}{3}}$ by $16^{- \frac{5}{6}}$ 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.

${(16^{\frac{5}{3} - \frac{5}{6}})}^{\frac{9}{5}}$

Step 2.2

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(16^{\frac{5}{3} \cdot \frac{2}{2} - \frac{5}{6}})}^{\frac{9}{5}}$

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 $\frac{5}{3}$ by $\frac{2}{2}$ .

${(16^{\frac{5 \cdot 2}{3 \cdot 2} - \frac{5}{6}})}^{\frac{9}{5}}$

Step 2.3.2

Multiply $3$ by $2$ .

${(16^{\frac{5 \cdot 2}{6} - \frac{5}{6}})}^{\frac{9}{5}}$

Step 2.4

Combine the numerators over the common denominator.

${(16^{\frac{5 \cdot 2 - 5}{6}})}^{\frac{9}{5}}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $5$ by $2$ .

${(16^{\frac{10 - 5}{6}})}^{\frac{9}{5}}$

Step 2.5.2

Subtract $5$ from $10$ .

${(16^{\frac{5}{6}})}^{\frac{9}{5}}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the exponents in ${(16^{\frac{5}{6}})}^{\frac{9}{5}}$ .

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Step 3.1.1

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

$16^{\frac{5}{6} \cdot \frac{9}{5}}$

Step 3.1.2

Cancel the common factor of $5$ .

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Step 3.1.2.1

Cancel the common factor.

$16^{\frac{5}{6} \cdot \frac{9}{5}}$

Step 3.1.2.2

Rewrite the expression.

$16^{\frac{1}{6} \cdot 9}$

Step 3.1.3

Cancel the common factor of $3$ .

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Step 3.1.3.1

Factor $3$ out of $6$ .

$16^{\frac{1}{3 (2)} \cdot 9}$

Step 3.1.3.2

Factor $3$ out of $9$ .

$16^{\frac{1}{3 \cdot 2} \cdot (3 \cdot 3)}$

Step 3.1.3.3

Cancel the common factor.

$16^{\frac{1}{3 \cdot 2} \cdot (3 \cdot 3)}$

Step 3.1.3.4

Rewrite the expression.

$16^{\frac{1}{2} \cdot 3}$

Step 3.1.4

Combine $\frac{1}{2}$ and $3$ .

$16^{\frac{3}{2}}$

Step 3.2

Rewrite $16$ as $4^{2}$ .

${(4^{2})}^{\frac{3}{2}}$

Step 3.3

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

$4^{2 (\frac{3}{2})}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

$4^{2 (\frac{3}{2})}$

Step 4.2

Rewrite the expression.

$4^{3}$

Step 5

Raise $4$ to the power of $3$ .

$64$