Pre-Algebra Examples

${(k + \frac{1}{2})}^{2} = \frac{9}{16}$

Step 1

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

$k + \frac{1}{2} = \pm \sqrt{\frac{9}{16}}$

Step 2

Simplify $\pm \sqrt{\frac{9}{16}}$ .

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Step 2.1

Rewrite $\sqrt{\frac{9}{16}}$ as $\frac{\sqrt{9}}{\sqrt{16}}$ .

$k + \frac{1}{2} = \pm \frac{\sqrt{9}}{\sqrt{16}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$k + \frac{1}{2} = \pm \frac{\sqrt{3^{2}}}{\sqrt{16}}$

Step 2.2.2

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

$k + \frac{1}{2} = \pm \frac{3}{\sqrt{16}}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Rewrite $16$ as $4^{2}$ .

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

Step 2.3.2

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

$k + \frac{1}{2} = \pm \frac{3}{4}$

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.

$k + \frac{1}{2} = \frac{3}{4}$

Step 3.2

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

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Step 3.2.1

Subtract $\frac{1}{2}$ from both sides of the equation.

$k = \frac{3}{4} - \frac{1}{2}$

Step 3.2.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$k = \frac{3}{4} - \frac{1}{2} \cdot \frac{2}{2}$

Step 3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$k = \frac{3}{4} - \frac{2}{2 \cdot 2}$

Step 3.2.3.2

Multiply $2$ by $2$ .

$k = \frac{3}{4} - \frac{2}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$k = \frac{3 - 2}{4}$

Step 3.2.5

Subtract $2$ from $3$ .

$k = \frac{1}{4}$

Step 3.3

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

$k + \frac{1}{2} = - \frac{3}{4}$

Step 3.4

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

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Step 3.4.1

Subtract $\frac{1}{2}$ from both sides of the equation.

$k = - \frac{3}{4} - \frac{1}{2}$

Step 3.4.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$k = - \frac{3}{4} - \frac{1}{2} \cdot \frac{2}{2}$

Step 3.4.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$k = - \frac{3}{4} - \frac{2}{2 \cdot 2}$

Step 3.4.3.2

Multiply $2$ by $2$ .

$k = - \frac{3}{4} - \frac{2}{4}$

Step 3.4.4

Combine the numerators over the common denominator.

$k = \frac{- 3 - 2}{4}$

Step 3.4.5

Subtract $2$ from $- 3$ .

$k = \frac{- 5}{4}$

Step 3.4.6

Move the negative in front of the fraction.

$k = - \frac{5}{4}$

Step 3.5

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

$k = \frac{1}{4}, - \frac{5}{4}$