Algebra Examples

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| {(x - 5)}^{\frac{1}{2}} | = | - 3 |$

Step 2

Solve for $x$ .

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Step 2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

${(x - 5)}^{\frac{1}{2}} = \pm | - 3 |$

Step 2.2

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

${(x - 5)}^{\frac{1}{2}} = \pm 3$

Step 2.3

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

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Step 2.3.1

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

${(x - 5)}^{\frac{1}{2}} = 3$

Step 2.3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(x - 5)}^{\frac{1}{2}})}^{2} = 3^{2}$

Step 2.3.3

Simplify the exponent.

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Step 2.3.3.1

Simplify the left side.

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Step 2.3.3.1.1

Simplify ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.3.1.1.1

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.3.1.1.1.1

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

${(x - 5)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 2.3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.3.1.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 2.3.3.1.1.1.2.2

Rewrite the expression.

${(x - 5)}^{1} = 3^{2}$

Step 2.3.3.1.1.2

Simplify.

$x - 5 = 3^{2}$

Step 2.3.3.2

Simplify the right side.

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Step 2.3.3.2.1

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

$x - 5 = 9$

Step 2.3.4

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

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Step 2.3.4.1

Add $5$ to both sides of the equation.

$x = 9 + 5$

Step 2.3.4.2

Add $9$ and $5$ .

$x = 14$

Step 2.3.5

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

${(x - 5)}^{\frac{1}{2}} = - 3$

Step 2.3.6

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 2.3.7

Simplify the exponent.

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Step 2.3.7.1

Simplify the left side.

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Step 2.3.7.1.1

Simplify ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.7.1.1.1

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.7.1.1.1.1

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

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 2.3.7.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.7.1.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 2.3.7.1.1.1.2.2

Rewrite the expression.

${(x - 5)}^{1} = {(- 3)}^{2}$

Step 2.3.7.1.1.2

Simplify.

$x - 5 = {(- 3)}^{2}$

Step 2.3.7.2

Simplify the right side.

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Step 2.3.7.2.1

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

$x - 5 = 9$

Step 2.3.8

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

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Step 2.3.8.1

Add $5$ to both sides of the equation.

$x = 9 + 5$

Step 2.3.8.2

Add $9$ and $5$ .

$x = 14$