Pre-Algebra Examples

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

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

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

$x$