Simplify ((16^(5/3))/(16^(5/6)))^(9/5)

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

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.

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

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $5$ by $2$ .

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

Subtract $5$ from $10$ .

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

Step 3

Simplify the expression.

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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 $2$ .

Cancel the common factor.

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

Rewrite the expression.

$4^{3}$

Step 5

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

$64$

Simplify ((16^(5/3))/(16^(5/6)))^(9/5)

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