Algebra Examples

$\sqrt{x} = \sqrt{x - 8} + 2$

Step 1

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

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{x - 8} + 2)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(\sqrt{x - 8} + 2)}^{2}$ as $(\sqrt{x - 8} + 2) (\sqrt{x - 8} + 2)$ .

$x = (\sqrt{x - 8} + 2) (\sqrt{x - 8} + 2)$

Step 2.3.1.2

Expand $(\sqrt{x - 8} + 2) (\sqrt{x - 8} + 2)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$x = \sqrt{x - 8} (\sqrt{x - 8} + 2) + 2 (\sqrt{x - 8} + 2)$

Step 2.3.1.2.2

Apply the distributive property.

$x = \sqrt{x - 8} \sqrt{x - 8} + \sqrt{x - 8} \cdot 2 + 2 (\sqrt{x - 8} + 2)$

Step 2.3.1.2.3

Apply the distributive property.

$x = \sqrt{x - 8} \sqrt{x - 8} + \sqrt{x - 8} \cdot 2 + 2 \sqrt{x - 8} + 2 \cdot 2$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $\sqrt{x - 8} \sqrt{x - 8}$ .

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Step 2.3.1.3.1.1.1

Raise $\sqrt{x - 8}$ to the power of $1$ .

$x = {\sqrt{x - 8}}^{1} \sqrt{x - 8} + \sqrt{x - 8} \cdot 2 + 2 \sqrt{x - 8} + 2 \cdot 2$

Step 2.3.1.3.1.1.2

Raise $\sqrt{x - 8}$ to the power of $1$ .

$x = {\sqrt{x - 8}}^{1} {\sqrt{x - 8}}^{1} + \sqrt{x - 8} \cdot 2 + 2 \sqrt{x - 8} + 2 \cdot 2$

Step 2.3.1.3.1.1.3

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

$x = {\sqrt{x - 8}}^{1 + 1} + \sqrt{x - 8} \cdot 2 + 2 \sqrt{x - 8} + 2 \cdot 2$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

$x = {\sqrt{x - 8}}^{2} + \sqrt{x - 8} \cdot 2 + 2 \sqrt{x - 8} + 2 \cdot 2$

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

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

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

Step 2.3.1.3.1.2.2

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

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

Step 2.3.1.3.1.2.3

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

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

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

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

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

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

Step 2.3.1.3.1.2.5

Simplify.

$x = x - 8 + \sqrt{x - 8} \cdot 2 + 2 \sqrt{x - 8} + 2 \cdot 2$

Step 2.3.1.3.1.3

Move $2$ to the left of $\sqrt{x - 8}$ .

$x = x - 8 + 2 \cdot \sqrt{x - 8} + 2 \sqrt{x - 8} + 2 \cdot 2$

Step 2.3.1.3.1.4

Multiply $2$ by $2$ .

$x = x - 8 + 2 \sqrt{x - 8} + 2 \sqrt{x - 8} + 4$

Step 2.3.1.3.2

Add $- 8$ and $4$ .

$x = x - 4 + 2 \sqrt{x - 8} + 2 \sqrt{x - 8}$

Step 2.3.1.3.3

Add $2 \sqrt{x - 8}$ and $2 \sqrt{x - 8}$ .

$x = x - 4 + 4 \sqrt{x - 8}$

Step 3

Solve for $4 \sqrt{x - 8}$ .

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Step 3.1

Rewrite the equation as $x - 4 + 4 \sqrt{x - 8} = x$ .

$x - 4 + 4 \sqrt{x - 8} = x$

Step 3.2

Move all terms not containing $4 \sqrt{x - 8}$ to the right side of the equation.

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Step 3.2.1

Subtract $x$ from both sides of the equation.

$- 4 + 4 \sqrt{x - 8} = x - x$

Step 3.2.2

Add $4$ to both sides of the equation.

$4 \sqrt{x - 8} = x - x + 4$

Step 3.2.3

Subtract $x$ from $x$ .

$4 \sqrt{x - 8} = 0 + 4$

Step 3.2.4

Add $0$ and $4$ .

$4 \sqrt{x - 8} = 4$

Step 4

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

${(4 \sqrt{x - 8})}^{2} = 4^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${(4 {(x - 8)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(4 {(x - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $4 {(x - 8)}^{\frac{1}{2}}$ .

$4^{2} {({(x - 8)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 5.2.1.2

Raise $4$ to the power of $2$ .

$16 {({(x - 8)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$16 {(x - 8)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$16 {(x - 8)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$16 {(x - 8)}^{1} = 4^{2}$

Step 5.2.1.4

Simplify.

$16 (x - 8) = 4^{2}$

Step 5.2.1.5

Apply the distributive property.

$16 x + 16 \cdot - 8 = 4^{2}$

Step 5.2.1.6

Multiply $16$ by $- 8$ .

$16 x - 128 = 4^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Raise $4$ to the power of $2$ .

$16 x - 128 = 16$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Add $128$ to both sides of the equation.

$16 x = 16 + 128$

Step 6.1.2

Add $16$ and $128$ .

$16 x = 144$

Step 6.2

Divide each term in $16 x = 144$ by $16$ and simplify.

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Step 6.2.1

Divide each term in $16 x = 144$ by $16$ .

$\frac{16 x}{16} = \frac{144}{16}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $16$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{144}{16}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{144}{16}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $144$ by $16$ .

$x = 9$