Simplify (-3 square root of 14x^3)/(- square root of 21x)

$\frac{- 3 \sqrt{14 x^{3}}}{- \sqrt{21 x}}$

Step 1

Combine $\sqrt{14 x^{3}}$ and $\sqrt{21 x}$ into a single radical.

$\frac{- 3 \sqrt{\frac{14 x^{3}}{21 x}}}{- 1}$

Step 2

Cancel the common factor of $14$ and $21$ .

Factor $7$ out of $14 x^{3}$ .

$\frac{- 3 \sqrt{\frac{7 (2 x^{3})}{21 x}}}{- 1}$

Cancel the common factors.

Factor $7$ out of $21 x$ .

$\frac{- 3 \sqrt{\frac{7 (2 x^{3})}{7 (3 x)}}}{- 1}$

Cancel the common factor.

$\frac{- 3 \sqrt{\frac{7 (2 x^{3})}{7 (3 x)}}}{- 1}$

Rewrite the expression.

$\frac{- 3 \sqrt{\frac{2 x^{3}}{3 x}}}{- 1}$

Step 3

Cancel the common factor of $x^{3}$ and $x$ .

Factor $x$ out of $2 x^{3}$ .

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

Cancel the common factors.

Factor $x$ out of $3 x$ .

$\frac{- 3 \sqrt{\frac{x (2 x^{2})}{x \cdot 3}}}{- 1}$

Cancel the common factor.

$\frac{- 3 \sqrt{\frac{x (2 x^{2})}{x \cdot 3}}}{- 1}$

Rewrite the expression.

$\frac{- 3 \sqrt{\frac{2 x^{2}}{3}}}{- 1}$

Step 4

Simplify the expression.

Move the negative one from the denominator of $\frac{- 3 \sqrt{\frac{2 x^{2}}{3}}}{- 1}$ .

$- 1 \cdot (- 3 \sqrt{\frac{2 x^{2}}{3}})$

Rewrite $- 1 \cdot (- 3 \sqrt{\frac{2 x^{2}}{3}})$ as $- (- 3 \sqrt{\frac{2 x^{2}}{3}})$ .

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

Step 5

Rewrite $\sqrt{\frac{2 x^{2}}{3}}$ as $\frac{\sqrt{2 x^{2}}}{\sqrt{3}}$ .

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

Step 6

Simplify the numerator.

Reorder $2$ and $x^{2}$ .

$- (- 3 \frac{\sqrt{x^{2} \cdot 2}}{\sqrt{3}})$

Pull terms out from under the radical.

$- (- 3 \frac{x \sqrt{2}}{\sqrt{3}})$

Step 7

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

$- (- 3 (\frac{x \sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))$

Step 8

Combine and simplify the denominator.

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

$- (- 3 \frac{x \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

$- (- 3 \frac{x \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- (- 3 \frac{x \sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$- (- 3 \frac{x \sqrt{2} \sqrt{3}}{3})$

Step 9

Simplify the numerator.

Combine using the product rule for radicals.

$- (- 3 \frac{x \sqrt{3 \cdot 2}}{3})$

Multiply $3$ by $2$ .

$- (- 3 \frac{x \sqrt{6}}{3})$

Step 10

Reduce the expression by cancelling the common factors.

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

$- (3 (- 1) \frac{x \sqrt{6}}{3})$

Cancel the common factor.

$- (3 \cdot - 1 \frac{x \sqrt{6}}{3})$

Rewrite the expression.

$- (- 1 (x \sqrt{6}))$

Rewrite $- 1 (x \sqrt{6})$ as $- (x \sqrt{6})$ .

$- - (x \sqrt{6})$

Step 11

Multiply $- (- x \sqrt{6})$ .

Multiply $- 1$ by $- 1$ .

$1 (x \sqrt{6})$

Multiply $x$ by $1$ .

$x \sqrt{6}$

Simplify (-3 square root of 14x^3)/(- square root of 21x)

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