Simplify (5x^-1y^(1/2))/( square root of 25x^4y^2)

$\frac{5 x^{- 1} y^{\frac{1}{2}}}{\sqrt{25 x^{4} y^{2}}}$

Step 1

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

$\frac{5 y^{\frac{1}{2}}}{\sqrt{25 x^{4} y^{2}} x}$

Step 2

Simplify the denominator.

Rewrite $25 x^{4} y^{2}$ as ${(5 x^{2} y)}^{2}$ .

$\frac{5 y^{\frac{1}{2}}}{\sqrt{{(5 x^{2} y)}^{2}} x}$

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

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

Combine exponents.

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

$\frac{5 y^{\frac{1}{2}}}{5 (x^{1} x^{2}) y}$

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

$\frac{5 y^{\frac{1}{2}}}{5 x^{1 + 2} y}$

Add $1$ and $2$ .

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

Step 3

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

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

Step 4

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

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

$\frac{5}{5 x^{3} (y^{- \frac{1}{2}} y)}$

Multiply $y^{- \frac{1}{2}}$ by $y$ .

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

$\frac{5}{5 x^{3} (y^{- \frac{1}{2}} y^{1})}$

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

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

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

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $2$ .

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

Step 5

Cancel the common factor.

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

Step 6

Rewrite the expression.

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