Pre-Algebra Examples

$- x^{2} = - \frac{1}{7}$

Step 1

Divide each term in $- x^{2} = - \frac{1}{7}$ by $- 1$ and simplify.

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Step 1.1

Divide each term in $- x^{2} = - \frac{1}{7}$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- \frac{1}{7}}{- 1}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- \frac{1}{7}}{- 1}$

Step 1.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- \frac{1}{7}}{- 1}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{\frac{1}{7}}{1}$

Step 1.3.2

Divide $\frac{1}{7}$ by $1$ .

$x^{2} = \frac{1}{7}$

Step 2

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

$x = \pm \sqrt{\frac{1}{7}}$

Step 3

Simplify $\pm \sqrt{\frac{1}{7}}$ .

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Step 3.1

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{7}}$

Step 3.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{7}}$

Step 3.3

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 3.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 3.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{2}}$

Step 3.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 3.4.6.2

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

$x = \pm \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3

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

$x = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{7}}{7^{1}}$

Step 3.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{7}}{7}$

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.

$x = \frac{\sqrt{7}}{7}$

Step 4.2

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

$x = - \frac{\sqrt{7}}{7}$

Step 4.3

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

$x = \frac{\sqrt{7}}{7}, - \frac{\sqrt{7}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{7}}{7}, - \frac{\sqrt{7}}{7}$

Decimal Form:

$x = 0.37796447 \dots, - 0.37796447 \dots$