Algebra Examples

$\sqrt{x + 8} = \sqrt{3 x + 8}$

Step 1

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

${\sqrt{x + 8}}^{2} = {\sqrt{3 x + 8}}^{2}$

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 $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

${({(x + 8)}^{\frac{1}{2}})}^{2} = {\sqrt{3 x + 8}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x + 8)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 x + 8}}^{2}$

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.

${(x + 8)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 x + 8}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 8)}^{1} = {\sqrt{3 x + 8}}^{2}$

Step 2.2.1.2

Simplify.

$x + 8 = {\sqrt{3 x + 8}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{3 x + 8}}^{2}$ as $3 x + 8$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 8}$ as ${(3 x + 8)}^{\frac{1}{2}}$ .

$x + 8 = {({(3 x + 8)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$x + 8 = {(3 x + 8)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$x + 8 = {(3 x + 8)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$x + 8 = {(3 x + 8)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$x + 8 = {(3 x + 8)}^{1}$

Step 2.3.1.5

Simplify.

$x + 8 = 3 x + 8$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $3 x$ from both sides of the equation.

$x + 8 - 3 x = 8$

Step 3.1.2

Subtract $3 x$ from $x$ .

$- 2 x + 8 = 8$

Step 3.2

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

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Step 3.2.1

Subtract $8$ from both sides of the equation.

$- 2 x = 8 - 8$

Step 3.2.2

Subtract $8$ from $8$ .

$- 2 x = 0$

Step 3.3

Divide each term in $- 2 x = 0$ by $- 2$ and simplify.

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Step 3.3.1

Divide each term in $- 2 x = 0$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{0}{- 2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{0}{- 2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- 2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $- 2$ .

$x = 0$