Algebra Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Raise $\sqrt{21 (x - 1)}$ to the power of $1$ .

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

Step 2.2

Raise $\sqrt{21 (x - 1)}$ to the power of $1$ .

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

Step 2.3

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

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

Step 2.4

Add $1$ and $1$ .

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

Step 2.5

Rewrite ${\sqrt{21 (x - 1)}}^{2}$ as $21 (x - 1)$ .

Tap for more steps...

Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21 (x - 1)}$ as ${(21 (x - 1))}^{\frac{1}{2}}$ .

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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

Step 2.6

Reorder $7$ and $x^{2}$ .

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

Step 2.7

Pull terms out from under the radical.

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

Step 2.8

Multiply $x \sqrt{7} \sqrt{21 (x - 1)}$ .

Tap for more steps...

Step 2.8.1

Combine using the product rule for radicals.

$\frac{21 (x - 1)}{x \sqrt{21 (x - 1) \cdot 7}}$

Step 2.8.2

Multiply $7$ by $21$ .

$\frac{21 (x - 1)}{x \sqrt{147 (x - 1)}}$

Step 2.9

Rewrite $147 (x - 1)$ as $7^{2} \cdot (3 (x - 1))$ .

Tap for more steps...

Step 2.9.1

Factor $49$ out of $147$ .

$\frac{21 (x - 1)}{x \sqrt{49 (3) (x - 1)}}$

Step 2.9.2

Rewrite $49$ as $7^{2}$ .

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

Step 2.9.3

Add parentheses.

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

Step 2.10

Pull terms out from under the radical.

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

Step 2.11

Rewrite using the commutative property of multiplication.

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