Simplify (3x^(-1/2)*3x^(1/2)y^(-1/3))/(3y^(-7/4))

$\frac{3 x^{- \frac{1}{2}} \cdot 3 x^{\frac{1}{2}} y^{- \frac{1}{3}}}{3 y^{- \frac{7}{4}}}$

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

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

$\frac{3 (x^{\frac{1}{2}} x^{- \frac{1}{2}}) \cdot (3 y^{- \frac{1}{3}})}{3 y^{- \frac{7}{4}}}$

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

$\frac{3 x^{\frac{1}{2} - \frac{1}{2}} \cdot (3 y^{- \frac{1}{3}})}{3 y^{- \frac{7}{4}}}$

Combine the numerators over the common denominator.

$\frac{3 x^{\frac{1 - 1}{2}} \cdot (3 y^{- \frac{1}{3}})}{3 y^{- \frac{7}{4}}}$

Subtract $1$ from $1$ .

$\frac{3 x^{\frac{0}{2}} \cdot (3 y^{- \frac{1}{3}})}{3 y^{- \frac{7}{4}}}$

Divide $0$ by $2$ .

$\frac{3 x^{0} \cdot (3 y^{- \frac{1}{3}})}{3 y^{- \frac{7}{4}}}$

Simplify $3 x^{0} \cdot (3 y^{- \frac{1}{3}})$ .

$\frac{3 \cdot (3 y^{- \frac{1}{3}})}{3 y^{- \frac{7}{4}}}$

Move $y^{- \frac{7}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{3 \cdot (3 y^{- \frac{1}{3}}) y^{\frac{7}{4}}}{3}$

Multiply $y^{- \frac{1}{3}}$ by $y^{\frac{7}{4}}$ by adding the exponents.

Move $y^{\frac{7}{4}}$ .

$\frac{3 \cdot (3 (y^{\frac{7}{4}} y^{- \frac{1}{3}}))}{3}$

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

$\frac{3 \cdot (3 y^{\frac{7}{4} - \frac{1}{3}})}{3}$

To write $\frac{7}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 \cdot (3 y^{\frac{7}{4} \cdot \frac{3}{3} - \frac{1}{3}})}{3}$

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 \cdot (3 y^{\frac{7}{4} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{4}{4}})}{3}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$\frac{3 \cdot (3 y^{\frac{7 \cdot 3}{4 \cdot 3} - \frac{1}{3} \cdot \frac{4}{4}})}{3}$

Multiply $4$ by $3$ .

$\frac{3 \cdot (3 y^{\frac{7 \cdot 3}{12} - \frac{1}{3} \cdot \frac{4}{4}})}{3}$

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\frac{3 \cdot (3 y^{\frac{7 \cdot 3}{12} - \frac{4}{3 \cdot 4}})}{3}$

Multiply $3$ by $4$ .

$\frac{3 \cdot (3 y^{\frac{7 \cdot 3}{12} - \frac{4}{12}})}{3}$

Combine the numerators over the common denominator.

$\frac{3 \cdot (3 y^{\frac{7 \cdot 3 - 4}{12}})}{3}$

Simplify the numerator.

Multiply $7$ by $3$ .

$\frac{3 \cdot (3 y^{\frac{21 - 4}{12}})}{3}$

Subtract $4$ from $21$ .

$\frac{3 \cdot (3 y^{\frac{17}{12}})}{3}$

Cancel the common factor.

$\frac{3 \cdot (3 y^{\frac{17}{12}})}{3}$

Divide $3 y^{\frac{17}{12}}$ by $1$ .

$3 y^{\frac{17}{12}}$