Algebra Examples

$\frac{\sqrt{18 x}}{4} + \sqrt{\frac{2 x}{4}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $18 x$ as $3^{2} \cdot (2 x)$ .

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Step 1.1.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2) x}}{4} + \sqrt{\frac{2 x}{4}}$

Step 1.1.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 2 x}}{4} + \sqrt{\frac{2 x}{4}}$

Step 1.1.1.3

Add parentheses.

$\frac{\sqrt{3^{2} \cdot (2 x)}}{4} + \sqrt{\frac{2 x}{4}}$

Step 1.1.2

Pull terms out from under the radical.

$\frac{3 \sqrt{2 x}}{4} + \sqrt{\frac{2 x}{4}}$

Step 1.2

Reduce the expression $\frac{2 x}{4}$ by cancelling the common factors.

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Step 1.2.1

Factor $2$ out of $2 x$ .

$\frac{3 \sqrt{2 x}}{4} + \sqrt{\frac{2 (x)}{4}}$

Step 1.2.2

Factor $2$ out of $4$ .

$\frac{3 \sqrt{2 x}}{4} + \sqrt{\frac{2 x}{2 \cdot 2}}$

Step 1.2.3

Cancel the common factor.

$\frac{3 \sqrt{2 x}}{4} + \sqrt{\frac{2 x}{2 \cdot 2}}$

Step 1.2.4

Rewrite the expression.

$\frac{3 \sqrt{2 x}}{4} + \sqrt{\frac{x}{2}}$

Step 1.3

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

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x}}{\sqrt{2}}$

Step 1.4

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.5.2

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

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.5.3

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

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.5.4

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

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.5.5

Add $1$ and $1$ .

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.5.6

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

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Step 1.5.6.1

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

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.5.6.2

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

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.5.6.3

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

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.5.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{2^{1}}$

Step 1.5.6.5

Evaluate the exponent.

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x} \sqrt{2}}{2}$

Step 1.6

Combine using the product rule for radicals.

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{x \cdot 2}}{2}$

Step 2

Combine $\frac{3 \sqrt{2 x}}{4}$ and $\frac{\sqrt{x \cdot 2}}{2}$ using a common denominator.

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Step 2.1

Reorder $x$ and $2$ .

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{2 \cdot x}}{2}$

Step 2.2

To write $\frac{\sqrt{2 \cdot x}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{2 \cdot x}}{2} \cdot \frac{2}{2}$

Step 2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{\sqrt{2 \cdot x}}{2}$ by $\frac{2}{2}$ .

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{2 \cdot x} \cdot 2}{2 \cdot 2}$

Step 2.3.2

Multiply $2$ by $2$ .

$\frac{3 \sqrt{2 x}}{4} + \frac{\sqrt{2 \cdot x} \cdot 2}{4}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 \sqrt{2 x} + \sqrt{2 \cdot x} \cdot 2}{4}$

Step 3

Simplify the numerator.

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Step 3.1

Move $2$ to the left of $\sqrt{2 \cdot x}$ .

$\frac{3 \sqrt{2 x} + 2 \cdot \sqrt{2 \cdot x}}{4}$

Step 3.2

Add $3 \sqrt{2 x}$ and $2 \sqrt{2 \cdot x}$ .

$\frac{5 \sqrt{2 x}}{4}$