Algebra Examples

$2^{3} \sqrt{{(x + 3)}^{2}} - 1 = 7$

Step 1

Add $1$ to both sides of the equation.

$2^{3} \sqrt{{(x + 3)}^{2}} = 7 + 1$

Step 2

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

${(2^{3} \sqrt{{(x + 3)}^{2}})}^{2} = {(7 + 1)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(2^{3} {(x + 3)}^{\frac{2}{2}})}^{2} = {(7 + 1)}^{2}$

Step 3.2

Divide $2$ by $2$ .

${(2^{3} {(x + 3)}^{1})}^{2} = {(7 + 1)}^{2}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${(2^{3} {(x + 3)}^{1})}^{2}$ .

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Step 3.3.1.1

Raise $2$ to the power of $3$ .

${(8 {(x + 3)}^{1})}^{2} = {(7 + 1)}^{2}$

Step 3.3.1.2

Simplify.

${(8 (x + 3))}^{2} = {(7 + 1)}^{2}$

Step 3.3.1.3

Apply the distributive property.

${(8 x + 8 \cdot 3)}^{2} = {(7 + 1)}^{2}$

Step 3.3.1.4

Multiply $8$ by $3$ .

${(8 x + 24)}^{2} = {(7 + 1)}^{2}$

Step 3.4

Simplify the right side.

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Step 3.4.1

Simplify ${(7 + 1)}^{2}$ .

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Step 3.4.1.1

Add $7$ and $1$ .

${(8 x + 24)}^{2} = 8^{2}$

Step 3.4.1.2

Raise $8$ to the power of $2$ .

${(8 x + 24)}^{2} = 64$

Step 4

Solve for $x$ .

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Step 4.1

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

$8 x + 24 = \pm \sqrt{64}$

Step 4.2

Simplify $\pm \sqrt{64}$ .

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Step 4.2.1

Rewrite $64$ as $8^{2}$ .

$8 x + 24 = \pm \sqrt{8^{2}}$

Step 4.2.2

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

$8 x + 24 = \pm 8$

Step 4.3

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

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Step 4.3.1

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

$8 x + 24 = 8$

Step 4.3.2

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

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Step 4.3.2.1

Subtract $24$ from both sides of the equation.

$8 x = 8 - 24$

Step 4.3.2.2

Subtract $24$ from $8$ .

$8 x = - 16$

Step 4.3.3

Divide each term in $8 x = - 16$ by $8$ and simplify.

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Step 4.3.3.1

Divide each term in $8 x = - 16$ by $8$ .

$\frac{8 x}{8} = \frac{- 16}{8}$

Step 4.3.3.2

Simplify the left side.

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Step 4.3.3.2.1

Cancel the common factor of $8$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 16}{8}$

Step 4.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 16}{8}$

Step 4.3.3.3

Simplify the right side.

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Step 4.3.3.3.1

Divide $- 16$ by $8$ .

$x = - 2$

Step 4.3.4

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

$8 x + 24 = - 8$

Step 4.3.5

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

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Step 4.3.5.1

Subtract $24$ from both sides of the equation.

$8 x = - 8 - 24$

Step 4.3.5.2

Subtract $24$ from $- 8$ .

$8 x = - 32$

Step 4.3.6

Divide each term in $8 x = - 32$ by $8$ and simplify.

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Step 4.3.6.1

Divide each term in $8 x = - 32$ by $8$ .

$\frac{8 x}{8} = \frac{- 32}{8}$

Step 4.3.6.2

Simplify the left side.

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Step 4.3.6.2.1

Cancel the common factor of $8$ .

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Step 4.3.6.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 32}{8}$

Step 4.3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 32}{8}$

Step 4.3.6.3

Simplify the right side.

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Step 4.3.6.3.1

Divide $- 32$ by $8$ .

$x = - 4$

Step 4.3.7

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

$x = - 2, - 4$