Simplify (x^5y^(-9/2)z^3)/(xy^-4z^3)

$\frac{x^{5} y^{- \frac{9}{2}} z^{3}}{x y^{- 4} z^{3}}$

Step 1

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

$\frac{x^{5} z^{3}}{x y^{- 4} z^{3} y^{\frac{9}{2}}}$

Step 2

Multiply $y^{- 4}$ by $y^{\frac{9}{2}}$ by adding the exponents.

Move $y^{\frac{9}{2}}$ .

$\frac{x^{5} z^{3}}{x (y^{\frac{9}{2}} y^{- 4}) z^{3}}$

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

$\frac{x^{5} z^{3}}{x y^{\frac{9}{2} - 4} z^{3}}$

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

$\frac{x^{5} z^{3}}{x y^{\frac{9}{2} - 4 \cdot \frac{2}{2}} z^{3}}$

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

$\frac{x^{5} z^{3}}{x y^{\frac{9}{2} + \frac{- 4 \cdot 2}{2}} z^{3}}$

Combine the numerators over the common denominator.

$\frac{x^{5} z^{3}}{x y^{\frac{9 - 4 \cdot 2}{2}} z^{3}}$

Simplify the numerator.

Multiply $- 4$ by $2$ .

$\frac{x^{5} z^{3}}{x y^{\frac{9 - 8}{2}} z^{3}}$

Subtract $8$ from $9$ .

$\frac{x^{5} z^{3}}{x y^{\frac{1}{2}} z^{3}}$

Step 3

Factor $x$ out of $x^{5} z^{3}$ .

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

Step 4

Cancel the common factors.

Factor $x$ out of $x y^{\frac{1}{2}} z^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor.

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

Step 6

Rewrite the expression.

$\frac{x^{4}}{y^{\frac{1}{2}}}$