Simplify ( square root of 28(x-1))÷( square root of 8x^2)

$\sqrt{28 (x - 1)} \div \sqrt{8 x^{2}}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt{28 (x - 1)}}{\sqrt{8 x^{2}}}$

Step 2

Combine $\sqrt{28 (x - 1)}$ and $\sqrt{8 x^{2}}$ into a single radical.

$\sqrt{\frac{28 (x - 1)}{8 x^{2}}}$

Step 3

Reduce the expression $\frac{28 (x - 1)}{8 x^{2}}$ by cancelling the common factors.

Factor $4$ out of $28 (x - 1)$ .

$\sqrt{\frac{4 (7 (x - 1))}{8 x^{2}}}$

Factor $4$ out of $8 x^{2}$ .

$\sqrt{\frac{4 (7 (x - 1))}{4 (2 x^{2})}}$

Cancel the common factor.

$\sqrt{\frac{4 (7 (x - 1))}{4 (2 x^{2})}}$

Rewrite the expression.

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

Step 4

Rewrite $\frac{7 (x - 1)}{2 x^{2}}$ as ${(\frac{1}{x})}^{2} \frac{7 (x - 1)}{2}$ .

Factor the perfect power $1^{2}$ out of $7 (x - 1)$ .

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

Factor the perfect power $x^{2}$ out of $2 x^{2}$ .

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

Rearrange the fraction $\frac{1^{2} (7 (x - 1))}{x^{2} \cdot 2}$ .

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

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite $\sqrt{\frac{7 (x - 1)}{2}}$ as $\frac{\sqrt{7 (x - 1)}}{\sqrt{2}}$ .

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

Step 7

Multiply $\frac{\sqrt{7 (x - 1)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8

Combine and simplify the denominator.

Multiply $\frac{\sqrt{7 (x - 1)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 9

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $7$ .

$\frac{1}{x} \cdot \frac{\sqrt{14 (x - 1)}}{2}$

Step 10

Combine fractions.

Multiply $\frac{1}{x}$ by $\frac{\sqrt{14 (x - 1)}}{2}$ .

$\frac{\sqrt{14 (x - 1)}}{x \cdot 2}$

Move $2$ to the left of $x$ .

$\frac{\sqrt{14 (x - 1)}}{2 x}$

Simplify ( square root of 28(x-1))÷( square root of 8x^2)

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