Precalculus Examples

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

Step 1

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

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

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.

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

Step 2.3.1.4.2

Rewrite the expression.

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

Step 2.3.1.5

Simplify.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

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

$4 {(x + 2)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Step 4.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.2.1.3.2.1

Cancel the common factor.

$4 {(x + 2)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Step 4.2.1.3.2.2

Rewrite the expression.

$4 {(x + 2)}^{1} = {(x + 2)}^{2}$

Step 4.2.1.4

Simplify.

$4 (x + 2) = {(x + 2)}^{2}$

Step 4.2.1.5

Apply the distributive property.

$4 x + 4 \cdot 2 = {(x + 2)}^{2}$

Step 4.2.1.6

Multiply $4$ by $2$ .

$4 x + 8 = {(x + 2)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(x + 2)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$4 x + 8 = (x + 2) (x + 2)$

Step 4.3.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$4 x + 8 = x (x + 2) + 2 (x + 2)$

Step 4.3.1.2.2

Apply the distributive property.

$4 x + 8 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 4.3.1.2.3

Apply the distributive property.

$4 x + 8 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.1.3.1.2

Move $2$ to the left of $x$ .

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

Step 4.3.1.3.1.3

Multiply $2$ by $2$ .

$4 x + 8 = x^{2} + 2 x + 2 x + 4$

Step 4.3.1.3.2

Add $2 x$ and $2 x$ .

$4 x + 8 = x^{2} + 4 x + 4$

Step 5

Solve for $x$ .

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Step 5.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 4 x + 4 = 4 x + 8$

Step 5.2

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

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Step 5.2.1

Subtract $4 x$ from both sides of the equation.

$x^{2} + 4 x + 4 - 4 x = 8$

Step 5.2.2

Combine the opposite terms in $x^{2} + 4 x + 4 - 4 x$ .

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Step 5.2.2.1

Subtract $4 x$ from $4 x$ .

$x^{2} + 0 + 4 = 8$

Step 5.2.2.2

Add $x^{2}$ and $0$ .

$x^{2} + 4 = 8$

Step 5.3

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

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Step 5.3.1

Subtract $4$ from both sides of the equation.

$x^{2} = 8 - 4$

Step 5.3.2

Subtract $4$ from $8$ .

$x^{2} = 4$

Step 5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 5.5

Simplify $\pm \sqrt{4}$ .

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Step 5.5.1

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

$x = \pm \sqrt{2^{2}}$

Step 5.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$