Pre-Algebra Examples

$\frac{\sqrt{x + 3}}{\sqrt{x}}$

Step 1

Write $\frac{\sqrt{x + 3}}{\sqrt{x}}$ as an equation.

$y = \frac{\sqrt{x + 3}}{\sqrt{x}}$

Step 2

Simplify $\frac{\sqrt{x + 3}}{\sqrt{x}}$ .

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Step 2.1

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

$y = \frac{\sqrt{x + 3}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

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

$y = \frac{\sqrt{x + 3} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 2.2.2

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

$y = \frac{\sqrt{x + 3} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 2.2.3

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

$y = \frac{\sqrt{x + 3} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 2.2.4

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

$y = \frac{\sqrt{x + 3} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 2.2.5

Add $1$ and $1$ .

$y = \frac{\sqrt{x + 3} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 2.2.6

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

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Step 2.2.6.1

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

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

Step 2.2.6.2

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

$y = \frac{\sqrt{x + 3} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 2.2.6.3

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

$y = \frac{\sqrt{x + 3} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{x + 3} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 2.2.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{x + 3} \sqrt{x}}{x^{1}}$

Step 2.2.6.5

Simplify.

$y = \frac{\sqrt{x + 3} \sqrt{x}}{x}$

Step 2.3

Combine using the product rule for radicals.

$y = \frac{\sqrt{(x + 3) x}}{x}$

Step 2.4

Reorder factors in $\frac{\sqrt{(x + 3) x}}{x}$ .

$y = \frac{\sqrt{x (x + 3)}}{x}$

Step 3

The given equation $y = \frac{\sqrt{x (x + 3)}}{x}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$