Simplify ((3y^(1/3))^2)/(y^(1/6))

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

Step 1

Simplify the numerator.

Apply the product rule to $3 y^{\frac{1}{3}}$ .

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

Raise $3$ to the power of $2$ .

$\frac{9 {(y^{\frac{1}{3}})}^{2}}{y^{\frac{1}{6}}}$

Multiply the exponents in ${(y^{\frac{1}{3}})}^{2}$ .

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

$\frac{9 y^{\frac{1}{3} \cdot 2}}{y^{\frac{1}{6}}}$

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

$\frac{9 y^{\frac{2}{3}}}{y^{\frac{1}{6}}}$

Step 2

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

$9 y^{\frac{2}{3}} y^{- \frac{1}{6}}$

Step 3

Multiply $y^{\frac{2}{3}}$ by $y^{- \frac{1}{6}}$ by adding the exponents.

Move $y^{- \frac{1}{6}}$ .

$9 (y^{- \frac{1}{6}} y^{\frac{2}{3}})$

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

$9 y^{- \frac{1}{6} + \frac{2}{3}}$

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

$9 y^{- \frac{1}{6} + \frac{2}{3} \cdot \frac{2}{2}}$

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

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

$9 y^{- \frac{1}{6} + \frac{2 \cdot 2}{3 \cdot 2}}$

Multiply $3$ by $2$ .

$9 y^{- \frac{1}{6} + \frac{2 \cdot 2}{6}}$

Combine the numerators over the common denominator.

$9 y^{\frac{- 1 + 2 \cdot 2}{6}}$

Simplify the numerator.

Multiply $2$ by $2$ .

$9 y^{\frac{- 1 + 4}{6}}$

Add $- 1$ and $4$ .

$9 y^{\frac{3}{6}}$

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3$ .

$9 y^{\frac{3 (1)}{6}}$

Cancel the common factors.

Factor $3$ out of $6$ .

$9 y^{\frac{3 \cdot 1}{3 \cdot 2}}$

Cancel the common factor.

$9 y^{\frac{3 \cdot 1}{3 \cdot 2}}$

Rewrite the expression.

$9 y^{\frac{1}{2}}$