Algebra Examples

$\sqrt{s + 31} = \frac{9}{\sqrt{s + 31}}$

Step 1

Solve for $\sqrt{s + 31}$ .

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Step 1.1

Find the LCD of the terms in the equation.

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Step 1.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, \sqrt{s + 31}$

Step 1.1.2

The LCM of one and any expression is the expression.

$\sqrt{s + 31}$

Step 1.2

Multiply each term in $\sqrt{s + 31} = \frac{9}{\sqrt{s + 31}}$ by $\sqrt{s + 31}$ to eliminate the fractions.

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Step 1.2.1

Multiply each term in $\sqrt{s + 31} = \frac{9}{\sqrt{s + 31}}$ by $\sqrt{s + 31}$ .

$\sqrt{s + 31} \sqrt{s + 31} = \frac{9}{\sqrt{s + 31}} \sqrt{s + 31}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Multiply $\sqrt{s + 31}$ by $\sqrt{s + 31}$ .

${\sqrt{s + 31}}^{2} = \frac{9}{\sqrt{s + 31}} \sqrt{s + 31}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $\sqrt{s + 31}$ .

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Step 1.2.3.1.1

Cancel the common factor.

${\sqrt{s + 31}}^{2} = \frac{9}{\sqrt{s + 31}} \sqrt{s + 31}$

Step 1.2.3.1.2

Rewrite the expression.

${\sqrt{s + 31}}^{2} = 9$

Step 1.3

Solve the equation.

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Step 1.3.1

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

$\sqrt{s + 31} = \pm \sqrt{9}$

Step 1.3.2

Simplify $\pm \sqrt{9}$ .

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Step 1.3.2.1

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

$\sqrt{s + 31} = \pm \sqrt{3^{2}}$

Step 1.3.2.2

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

$\sqrt{s + 31} = \pm 3$

Step 1.3.3

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

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Step 1.3.3.1

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

$\sqrt{s + 31} = 3$

Step 1.3.3.2

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

$\sqrt{s + 31} = - 3$

Step 1.3.3.3

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

$\sqrt{s + 31} = 3, - 3$

Step 2

Solve for $s$ in $\sqrt{s + 31} = 3$ .

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Step 2.1

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

${\sqrt{s + 31}}^{2} = 3^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{s + 31}$ as ${(s + 31)}^{\frac{1}{2}}$ .

${({(s + 31)}^{\frac{1}{2}})}^{2} = 3^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(s + 31)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(s + 31)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(s + 31)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(s + 31)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(s + 31)}^{1} = 3^{2}$

Step 2.2.2.1.2

Simplify.

$s + 31 = 3^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

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

$s + 31 = 9$

Step 2.3

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

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Step 2.3.1

Subtract $31$ from both sides of the equation.

$s = 9 - 31$

Step 2.3.2

Subtract $31$ from $9$ .

$s = - 22$

Step 3

Solve for $s$ in $\sqrt{s + 31} = - 3$ .

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Step 3.1

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

${\sqrt{s + 31}}^{2} = {(- 3)}^{2}$

Step 3.2

Simplify each side of the equation.

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{s + 31}$ as ${(s + 31)}^{\frac{1}{2}}$ .

${({(s + 31)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify ${({(s + 31)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1

Multiply the exponents in ${({(s + 31)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1.1

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

${(s + 31)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.2.1

Cancel the common factor.

${(s + 31)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

${(s + 31)}^{1} = {(- 3)}^{2}$

Step 3.2.2.1.2

Simplify.

$s + 31 = {(- 3)}^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

$s + 31 = 9$

Step 3.3

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

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Step 3.3.1

Subtract $31$ from both sides of the equation.

$s = 9 - 31$

Step 3.3.2

Subtract $31$ from $9$ .

$s = - 22$

Step 4

List all of the solutions.

$s = - 22, - 22$