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

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the denominator.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $3$ .

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

Divide $2$ by $2$ .

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

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

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

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

Move $y^{2}$ .

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

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

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

Add $2$ and $3$ .

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

Step 4

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

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

Step 5

Multiply $z^{4}$ by $z^{- \frac{3}{2}}$ by adding the exponents.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $2$ .

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

Subtract $3$ from $8$ .

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

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

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