Algebra Examples

$\sqrt{x - 2} = \sqrt{12}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x - 2}}^{2} = {\sqrt{12}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(x - 2)}^{\frac{1}{2}})}^{2} = {\sqrt{12}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x - 2)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x - 2)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x - 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{12}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x - 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{12}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x - 2)}^{1} = {\sqrt{12}}^{2}$

Step 2.2.1.2

Simplify.

$x - 2 = {\sqrt{12}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{12}}^{2}$ .

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Factor $4$ out of $12$ .

$x - 2 = {\sqrt{4 (3)}}^{2}$

Step 2.3.1.1.2

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

$x - 2 = {\sqrt{2^{2} \cdot 3}}^{2}$

Step 2.3.1.2

Pull terms out from under the radical.

$x - 2 = {(2 \sqrt{3})}^{2}$

Step 2.3.1.3

Simplify the expression.

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Step 2.3.1.3.1

Apply the product rule to $2 \sqrt{3}$ .

$x - 2 = 2^{2} {\sqrt{3}}^{2}$

Step 2.3.1.3.2

Raise $2$ to the power of $2$ .

$x - 2 = 4 {\sqrt{3}}^{2}$

Step 2.3.1.4

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

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Step 2.3.1.4.1

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

$x - 2 = 4 {(3^{\frac{1}{2}})}^{2}$

Step 2.3.1.4.2

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

$x - 2 = 4 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 2.3.1.4.3

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

$x - 2 = 4 \cdot 3^{\frac{2}{2}}$

Step 2.3.1.4.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.4.1

Cancel the common factor.

$x - 2 = 4 \cdot 3^{\frac{2}{2}}$

Step 2.3.1.4.4.2

Rewrite the expression.

$x - 2 = 4 \cdot 3^{1}$

Step 2.3.1.4.5

Evaluate the exponent.

$x - 2 = 4 \cdot 3$

Step 2.3.1.5

Multiply $4$ by $3$ .

$x - 2 = 12$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Add $2$ to both sides of the equation.

$x = 12 + 2$

Step 3.2

Add $12$ and $2$ .

$x = 14$