Pre-Algebra Examples

$- 9 {(z + \frac{1}{3})}^{2} = - 49$

Step 1

Divide each term in $- 9 {(z + \frac{1}{3})}^{2} = - 49$ by $- 9$ and simplify.

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Step 1.1

Divide each term in $- 9 {(z + \frac{1}{3})}^{2} = - 49$ by $- 9$ .

$\frac{- 9 {(z + \frac{1}{3})}^{2}}{- 9} = \frac{- 49}{- 9}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 9 {(z + \frac{1}{3})}^{2}}{- 9} = \frac{- 49}{- 9}$

Step 1.2.1.2

Divide ${(z + \frac{1}{3})}^{2}$ by $1$ .

${(z + \frac{1}{3})}^{2} = \frac{- 49}{- 9}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Dividing two negative values results in a positive value.

${(z + \frac{1}{3})}^{2} = \frac{49}{9}$

Step 2

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

$z + \frac{1}{3} = \pm \sqrt{\frac{49}{9}}$

Step 3

Simplify $\pm \sqrt{\frac{49}{9}}$ .

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Step 3.1

Rewrite $\sqrt{\frac{49}{9}}$ as $\frac{\sqrt{49}}{\sqrt{9}}$ .

$z + \frac{1}{3} = \pm \frac{\sqrt{49}}{\sqrt{9}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $49$ as $7^{2}$ .

$z + \frac{1}{3} = \pm \frac{\sqrt{7^{2}}}{\sqrt{9}}$

Step 3.2.2

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

$z + \frac{1}{3} = \pm \frac{7}{\sqrt{9}}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

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

$z + \frac{1}{3} = \pm \frac{7}{\sqrt{3^{2}}}$

Step 3.3.2

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

$z + \frac{1}{3} = \pm \frac{7}{3}$

Step 4

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

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Step 4.1

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

$z + \frac{1}{3} = \frac{7}{3}$

Step 4.2

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

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Step 4.2.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$z = \frac{7}{3} - \frac{1}{3}$

Step 4.2.2

Combine the numerators over the common denominator.

$z = \frac{7 - 1}{3}$

Step 4.2.3

Subtract $1$ from $7$ .

$z = \frac{6}{3}$

Step 4.2.4

Divide $6$ by $3$ .

$z = 2$

Step 4.3

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

$z + \frac{1}{3} = - \frac{7}{3}$

Step 4.4

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

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Step 4.4.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$z = - \frac{7}{3} - \frac{1}{3}$

Step 4.4.2

Combine the numerators over the common denominator.

$z = \frac{- 7 - 1}{3}$

Step 4.4.3

Subtract $1$ from $- 7$ .

$z = \frac{- 8}{3}$

Step 4.4.4

Move the negative in front of the fraction.

$z = - \frac{8}{3}$

Step 4.5

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

$z = 2, - \frac{8}{3}$