Pre-Algebra Examples

${(\frac{1}{3} m + 4)}^{2} = 27$

Step 1

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

$\frac{1}{3} m + 4 = \pm \sqrt{27}$

Step 2

Simplify $\pm \sqrt{27}$ .

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Step 2.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.1.1

Factor $9$ out of $27$ .

$\frac{1}{3} m + 4 = \pm \sqrt{9 (3)}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{1}{3} m + 4 = \pm \sqrt{3^{2} \cdot 3}$

Step 2.2

Pull terms out from under the radical.

$\frac{1}{3} m + 4 = \pm 3 \sqrt{3}$

Step 3

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

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Step 3.1

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

$\frac{1}{3} m + 4 = 3 \sqrt{3}$

Step 3.2

Combine $\frac{1}{3}$ and $m$ .

$\frac{m}{3} + 4 = 3 \sqrt{3}$

Step 3.3

Subtract $4$ from both sides of the equation.

$\frac{m}{3} = 3 \sqrt{3} - 4$

Step 3.4

Multiply both sides of the equation by $3$ .

$3 \frac{m}{3} = 3 (3 \sqrt{3} - 4)$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Cancel the common factor of $3$ .

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Step 3.5.1.1.1

Cancel the common factor.

$3 \frac{m}{3} = 3 (3 \sqrt{3} - 4)$

Step 3.5.1.1.2

Rewrite the expression.

$m = 3 (3 \sqrt{3} - 4)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $3 (3 \sqrt{3} - 4)$ .

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Step 3.5.2.1.1

Apply the distributive property.

$m = 3 (3 \sqrt{3}) + 3 \cdot - 4$

Step 3.5.2.1.2

Multiply.

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Step 3.5.2.1.2.1

Multiply $3$ by $3$ .

$m = 9 \sqrt{3} + 3 \cdot - 4$

Step 3.5.2.1.2.2

Multiply $3$ by $- 4$ .

$m = 9 \sqrt{3} - 12$

Step 3.6

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

$\frac{1}{3} m + 4 = - 3 \sqrt{3}$

Step 3.7

Combine $\frac{1}{3}$ and $m$ .

$\frac{m}{3} + 4 = - 3 \sqrt{3}$

Step 3.8

Subtract $4$ from both sides of the equation.

$\frac{m}{3} = - 3 \sqrt{3} - 4$

Step 3.9

Multiply both sides of the equation by $3$ .

$3 \frac{m}{3} = 3 (- 3 \sqrt{3} - 4)$

Step 3.10

Simplify both sides of the equation.

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Step 3.10.1

Simplify the left side.

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Step 3.10.1.1

Cancel the common factor of $3$ .

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Step 3.10.1.1.1

Cancel the common factor.

$3 \frac{m}{3} = 3 (- 3 \sqrt{3} - 4)$

Step 3.10.1.1.2

Rewrite the expression.

$m = 3 (- 3 \sqrt{3} - 4)$

Step 3.10.2

Simplify the right side.

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Step 3.10.2.1

Simplify $3 (- 3 \sqrt{3} - 4)$ .

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Step 3.10.2.1.1

Apply the distributive property.

$m = 3 (- 3 \sqrt{3}) + 3 \cdot - 4$

Step 3.10.2.1.2

Multiply.

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Step 3.10.2.1.2.1

Multiply $- 3$ by $3$ .

$m = - 9 \sqrt{3} + 3 \cdot - 4$

Step 3.10.2.1.2.2

Multiply $3$ by $- 4$ .

$m = - 9 \sqrt{3} - 12$

Step 3.11

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

$m = 9 \sqrt{3} - 12, - 9 \sqrt{3} - 12$

Step 4

The result can be shown in multiple forms.

Exact Form:

$m = 9 \sqrt{3} - 12, - 9 \sqrt{3} - 12$

Decimal Form:

$m = 3.58845726 \dots, - 27.58845726 \dots$