Algebra Examples

$\sqrt{3} x + \sqrt{12} + \frac{x + 5}{\sqrt{3}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1

Factor $4$ out of $12$ .

$\sqrt{3} x + \sqrt{4 (3)} + \frac{x + 5}{\sqrt{3}}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{3} x + \sqrt{2^{2} \cdot 3} + \frac{x + 5}{\sqrt{3}}$

Step 1.2

Pull terms out from under the radical.

$\sqrt{3} x + 2 \sqrt{3} + \frac{x + 5}{\sqrt{3}}$

Step 1.3

Multiply $\frac{x + 5}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{3} x + 2 \sqrt{3} + \frac{x + 5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{x + 5}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.4.4

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

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.4.5

Add $1$ and $1$ .

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.4.6.2

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

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.4.6.3

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

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.4.6.4.2

Rewrite the expression.

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{3^{1}}$

Step 1.4.6.5

Evaluate the exponent.

$\sqrt{3} x + 2 \sqrt{3} + \frac{(x + 5) \sqrt{3}}{3}$

Step 2

To write $\sqrt{3} x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{3} x \cdot \frac{3}{3} + \frac{(x + 5) \sqrt{3}}{3} + 2 \sqrt{3}$

Step 3

Simplify terms.

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Step 3.1

Combine $\sqrt{3} x$ and $\frac{3}{3}$ .

$\frac{\sqrt{3} x \cdot 3}{3} + \frac{(x + 5) \sqrt{3}}{3} + 2 \sqrt{3}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\sqrt{3} x \cdot 3 + (x + 5) \sqrt{3}}{3} + 2 \sqrt{3}$

Step 4

Simplify the numerator.

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Step 4.1

Move $3$ to the left of $\sqrt{3} x$ .

$\frac{3 \cdot (\sqrt{3} x) + (x + 5) \sqrt{3}}{3} + 2 \sqrt{3}$

Step 4.2

Apply the distributive property.

$\frac{3 \sqrt{3} x + x \sqrt{3} + 5 \sqrt{3}}{3} + 2 \sqrt{3}$

Step 4.3

Reorder the factors of $3 \sqrt{3} x$ .

$\frac{3 x \sqrt{3} + x \sqrt{3} + 5 \sqrt{3}}{3} + 2 \sqrt{3}$

Step 4.4

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

$\frac{4 x \sqrt{3} + 5 \sqrt{3}}{3} + 2 \sqrt{3}$