Simplify (3y^(1/4))/(4x^(-2/3)y^(3/2)*3y^(1/2))

$\frac{3 y^{\frac{1}{4}}}{4 x^{- \frac{2}{3}} y^{\frac{3}{2}} \cdot 3 y^{\frac{1}{2}}}$

Step 1

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

Move $y^{\frac{1}{2}}$ .

$\frac{3 y^{\frac{1}{4}}}{4 x^{- \frac{2}{3}} (y^{\frac{1}{2}} y^{\frac{3}{2}}) \cdot 3}$

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

$\frac{3 y^{\frac{1}{4}}}{4 x^{- \frac{2}{3}} y^{\frac{1}{2} + \frac{3}{2}} \cdot 3}$

Combine the numerators over the common denominator.

$\frac{3 y^{\frac{1}{4}}}{4 x^{- \frac{2}{3}} y^{\frac{1 + 3}{2}} \cdot 3}$

Add $1$ and $3$ .

$\frac{3 y^{\frac{1}{4}}}{4 x^{- \frac{2}{3}} y^{\frac{4}{2}} \cdot 3}$

Divide $4$ by $2$ .

$\frac{3 y^{\frac{1}{4}}}{4 x^{- \frac{2}{3}} y^{2} \cdot 3}$

Step 2

Move $y^{\frac{1}{4}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{3}{4 x^{- \frac{2}{3}} y^{2} \cdot 3 y^{- \frac{1}{4}}}$

Step 3

Multiply $y^{2}$ by $y^{- \frac{1}{4}}$ by adding the exponents.

Move $y^{- \frac{1}{4}}$ .

$\frac{3}{4 x^{- \frac{2}{3}} (y^{- \frac{1}{4}} y^{2}) \cdot 3}$

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

$\frac{3}{4 x^{- \frac{2}{3}} y^{- \frac{1}{4} + 2} \cdot 3}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3}{4 x^{- \frac{2}{3}} y^{- \frac{1}{4} + 2 \cdot \frac{4}{4}} \cdot 3}$

Combine $2$ and $\frac{4}{4}$ .

$\frac{3}{4 x^{- \frac{2}{3}} y^{- \frac{1}{4} + \frac{2 \cdot 4}{4}} \cdot 3}$

Combine the numerators over the common denominator.

$\frac{3}{4 x^{- \frac{2}{3}} y^{\frac{- 1 + 2 \cdot 4}{4}} \cdot 3}$

Simplify the numerator.

Multiply $2$ by $4$ .

$\frac{3}{4 x^{- \frac{2}{3}} y^{\frac{- 1 + 8}{4}} \cdot 3}$

Add $- 1$ and $8$ .

$\frac{3}{4 x^{- \frac{2}{3}} y^{\frac{7}{4}} \cdot 3}$

Step 4

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

$\frac{3 x^{\frac{2}{3}}}{4 y^{\frac{7}{4}} \cdot 3}$

Step 5

Cancel the common factor.

$\frac{3 x^{\frac{2}{3}}}{4 y^{\frac{7}{4}} \cdot 3}$

Step 6

Rewrite the expression.

$\frac{x^{\frac{2}{3}}}{4 y^{\frac{7}{4}}}$