Algebra Examples

$\sqrt{x^{2}} = 1$

Step 1

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

${\sqrt{x^{2}}}^{2} = 1^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

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

Step 2.2

Divide $2$ by $2$ .

${(x^{1})}^{2} = 1^{2}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Multiply the exponents in ${(x^{1})}^{2}$ .

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Step 2.3.1.1

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

$x^{1 \cdot 2} = 1^{2}$

Step 2.3.1.2

Multiply $2$ by $1$ .

$x^{2} = 1^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

One to any power is one.

$x^{2} = 1$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \pm \sqrt{1}$

Step 3.2

Any root of $1$ is $1$ .

$x = \pm 1$

Step 3.3

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

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Step 3.3.1

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

$x = 1$

Step 3.3.2

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

$x = - 1$

Step 3.3.3

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

$x = 1, - 1$