Simplify ( square root of 200x^3)/( square root of 10x^-1)

$\frac{\sqrt{200 x^{3}}}{\sqrt{10 x^{- 1}}}$

Step 1

Combine $\sqrt{200 x^{3}}$ and $\sqrt{10 x^{- 1}}$ into a single radical.

$\sqrt{\frac{200 x^{3}}{10 x^{- 1}}}$

Step 2

Reduce the expression $\frac{200 x^{3}}{10 x^{- 1}}$ by cancelling the common factors.

Factor $10$ out of $200 x^{3}$ .

$\sqrt{\frac{10 (20 x^{3})}{10 x^{- 1}}}$

Factor $10$ out of $10 x^{- 1}$ .

$\sqrt{\frac{10 (20 x^{3})}{10 (x^{- 1})}}$

Cancel the common factor.

$\sqrt{\frac{10 (20 x^{3})}{10 x^{- 1}}}$

Rewrite the expression.

$\sqrt{\frac{20 x^{3}}{x^{- 1}}}$

Step 3

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

$\sqrt{20 x^{3} x}$

Step 4

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

Move $x$ .

$\sqrt{20 (x \cdot x^{3})}$

Multiply $x$ by $x^{3}$ .

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

$\sqrt{20 (x^{1} x^{3})}$

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

$\sqrt{20 x^{1 + 3}}$

Add $1$ and $3$ .

$\sqrt{20 x^{4}}$

Step 5

Rewrite $20 x^{4}$ as ${(2 x^{2})}^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\sqrt{4 (5) x^{4}}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 5 x^{4}}$

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

$\sqrt{2^{2} \cdot 5 {(x^{2})}^{2}}$

Move $5$ .

$\sqrt{2^{2} {(x^{2})}^{2} \cdot 5}$

Rewrite $2^{2} {(x^{2})}^{2}$ as ${(2 x^{2})}^{2}$ .

$\sqrt{{(2 x^{2})}^{2} \cdot 5}$

Step 6

Pull terms out from under the radical.

$2 x^{2} \sqrt{5}$