Simplify (x^(3/2)y^(7/3))/(x^3y^(2/9))

$\frac{x^{\frac{3}{2}} y^{\frac{7}{3}}}{x^{3} y^{\frac{2}{9}}}$

Step 1

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

$\frac{y^{\frac{7}{3}}}{x^{3} y^{\frac{2}{9}} x^{- \frac{3}{2}}}$

Step 2

Multiply $x^{3}$ by $x^{- \frac{3}{2}}$ by adding the exponents.

Move $x^{- \frac{3}{2}}$ .

$\frac{y^{\frac{7}{3}}}{x^{- \frac{3}{2}} x^{3} y^{\frac{2}{9}}}$

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

$\frac{y^{\frac{7}{3}}}{x^{- \frac{3}{2} + 3} y^{\frac{2}{9}}}$

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

$\frac{y^{\frac{7}{3}}}{x^{- \frac{3}{2} + 3 \cdot \frac{2}{2}} y^{\frac{2}{9}}}$

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

$\frac{y^{\frac{7}{3}}}{x^{- \frac{3}{2} + \frac{3 \cdot 2}{2}} y^{\frac{2}{9}}}$

Combine the numerators over the common denominator.

$\frac{y^{\frac{7}{3}}}{x^{\frac{- 3 + 3 \cdot 2}{2}} y^{\frac{2}{9}}}$

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{y^{\frac{7}{3}}}{x^{\frac{- 3 + 6}{2}} y^{\frac{2}{9}}}$

Add $- 3$ and $6$ .

$\frac{y^{\frac{7}{3}}}{x^{\frac{3}{2}} y^{\frac{2}{9}}}$

Step 3

Move $y^{\frac{2}{9}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{y^{\frac{7}{3}} y^{- \frac{2}{9}}}{x^{\frac{3}{2}}}$

Step 4

Multiply $y^{\frac{7}{3}}$ by $y^{- \frac{2}{9}}$ by adding the exponents.

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

$\frac{y^{\frac{7}{3} - \frac{2}{9}}}{x^{\frac{3}{2}}}$

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

$\frac{y^{\frac{7}{3} \cdot \frac{3}{3} - \frac{2}{9}}}{x^{\frac{3}{2}}}$

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{y^{\frac{7 \cdot 3}{3 \cdot 3} - \frac{2}{9}}}{x^{\frac{3}{2}}}$

Multiply $3$ by $3$ .

$\frac{y^{\frac{7 \cdot 3}{9} - \frac{2}{9}}}{x^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{y^{\frac{7 \cdot 3 - 2}{9}}}{x^{\frac{3}{2}}}$

Simplify the numerator.

Multiply $7$ by $3$ .

$\frac{y^{\frac{21 - 2}{9}}}{x^{\frac{3}{2}}}$

Subtract $2$ from $21$ .

$\frac{y^{\frac{19}{9}}}{x^{\frac{3}{2}}}$