Simplify ( square root of 9x^2y^4)/(4x^(-3/2)y^4)

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

Step 1

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

$\frac{\sqrt{9 x^{2} y^{4}} x^{\frac{3}{2}}}{4 y^{4}}$

Step 2

Simplify the numerator.

Rewrite $9 x^{2} y^{4}$ as ${(3 x y^{2})}^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

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

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

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

Multiply $x^{\frac{3}{2}}$ by $x$ .

Raise $x$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $3$ and $2$ .

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

Step 3

Factor $y^{2}$ out of $3 x^{\frac{5}{2}} y^{2}$ .

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

Step 4

Cancel the common factors.

Factor $y^{2}$ out of $4 y^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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