Evaluate ((6^(1/5))/(6^(2/5)))^3

${(\frac{6^{\frac{1}{5}}}{6^{\frac{2}{5}}})}^{3}$

Step 1

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

${(\frac{1}{6^{\frac{2}{5}} \cdot 6^{- \frac{1}{5}}})}^{3}$

Step 2

Multiply $6^{\frac{2}{5}}$ by $6^{- \frac{1}{5}}$ by adding the exponents.

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

${(\frac{1}{6^{\frac{2}{5} - \frac{1}{5}}})}^{3}$

Combine the numerators over the common denominator.

${(\frac{1}{6^{\frac{2 - 1}{5}}})}^{3}$

Subtract $1$ from $2$ .

${(\frac{1}{6^{\frac{1}{5}}})}^{3}$

Step 3

Apply the product rule to $\frac{1}{6^{\frac{1}{5}}}$ .

$\frac{1^{3}}{{(6^{\frac{1}{5}})}^{3}}$

Step 4

One to any power is one.

$\frac{1}{{(6^{\frac{1}{5}})}^{3}}$

Step 5

Multiply the exponents in ${(6^{\frac{1}{5}})}^{3}$ .

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

$\frac{1}{6^{\frac{1}{5} \cdot 3}}$

Combine $\frac{1}{5}$ and $3$ .

$\frac{1}{6^{\frac{3}{5}}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6^{\frac{3}{5}}}$

Decimal Form:

$0.34127875 \dots$