Algebra Examples

$\frac{\sqrt{20 x + 5}}{\sqrt{5 x}}$

Step 1

Combine $\sqrt{20 x + 5}$ and $\sqrt{5 x}$ into a single radical.

$\sqrt{\frac{20 x + 5}{5 x}}$

Step 2

Factor $5$ out of $20 x + 5$ .

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Step 2.1

Factor $5$ out of $20 x$ .

$\sqrt{\frac{5 (4 x) + 5}{5 x}}$

Step 2.2

Factor $5$ out of $5$ .

$\sqrt{\frac{5 (4 x) + 5 (1)}{5 x}}$

Step 2.3

Factor $5$ out of $5 (4 x) + 5 (1)$ .

$\sqrt{\frac{5 (4 x + 1)}{5 x}}$

Step 3

Reduce the expression $\frac{5 (4 x + 1)}{5 x}$ by cancelling the common factors.

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Step 3.1

Cancel the common factor.

$\sqrt{\frac{5 (4 x + 1)}{5 x}}$

Step 3.2

Rewrite the expression.

$\sqrt{\frac{4 x + 1}{x}}$

Step 4

Rewrite $\sqrt{\frac{4 x + 1}{x}}$ as $\frac{\sqrt{4 x + 1}}{\sqrt{x}}$ .

$\frac{\sqrt{4 x + 1}}{\sqrt{x}}$

Step 5

Multiply $\frac{\sqrt{4 x + 1}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{4 x + 1}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{4 x + 1}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{4 x + 1} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 6.2

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

$\frac{\sqrt{4 x + 1} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 6.3

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

$\frac{\sqrt{4 x + 1} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 6.4

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

$\frac{\sqrt{4 x + 1} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{\sqrt{4 x + 1} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

$\frac{\sqrt{4 x + 1} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{\sqrt{4 x + 1} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{\sqrt{4 x + 1} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{\sqrt{4 x + 1} \sqrt{x}}{x^{1}}$

Step 6.6.5

Simplify.

$\frac{\sqrt{4 x + 1} \sqrt{x}}{x}$

Step 7

Combine using the product rule for radicals.

$\frac{\sqrt{(4 x + 1) x}}{x}$

Step 8

Reorder factors in $\frac{\sqrt{(4 x + 1) x}}{x}$ .

$\frac{\sqrt{x (4 x + 1)}}{x}$