Algebra Examples

$\sqrt{x^{2}} = \sqrt{40}$

Step 1

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

${\sqrt{x^{2}}}^{2} = {\sqrt{40}}^{2}$

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 $x^{\frac{2}{2}}$ .

${(x^{\frac{2}{2}})}^{2} = {\sqrt{40}}^{2}$

Step 2.2

Divide $2$ by $2$ .

${(x^{1})}^{2} = {\sqrt{40}}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{1 \cdot 2} = {\sqrt{40}}^{2}$

Step 2.3.1.2

Multiply $2$ by $1$ .

$x^{2} = {\sqrt{40}}^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify ${\sqrt{40}}^{2}$ .

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Step 2.4.1.1

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 2.4.1.1.1

Factor $4$ out of $40$ .

$x^{2} = {\sqrt{4 (10)}}^{2}$

Step 2.4.1.1.2

Rewrite $4$ as $2^{2}$ .

$x^{2} = {\sqrt{2^{2} \cdot 10}}^{2}$

Step 2.4.1.2

Pull terms out from under the radical.

$x^{2} = {(2 \sqrt{10})}^{2}$

Step 2.4.1.3

Simplify the expression.

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Step 2.4.1.3.1

Apply the product rule to $2 \sqrt{10}$ .

$x^{2} = 2^{2} {\sqrt{10}}^{2}$

Step 2.4.1.3.2

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

$x^{2} = 4 {\sqrt{10}}^{2}$

Step 2.4.1.4

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$x^{2} = 4 {(10^{\frac{1}{2}})}^{2}$

Step 2.4.1.4.2

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

$x^{2} = 4 \cdot 10^{\frac{1}{2} \cdot 2}$

Step 2.4.1.4.3

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

$x^{2} = 4 \cdot 10^{\frac{2}{2}}$

Step 2.4.1.4.4

Cancel the common factor of $2$ .

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Step 2.4.1.4.4.1

Cancel the common factor.

$x^{2} = 4 \cdot 10^{\frac{2}{2}}$

Step 2.4.1.4.4.2

Rewrite the expression.

$x^{2} = 4 \cdot 10^{1}$

Step 2.4.1.4.5

Evaluate the exponent.

$x^{2} = 4 \cdot 10$

Step 2.4.1.5

Multiply $4$ by $10$ .

$x^{2} = 40$

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.

$x = \pm \sqrt{40}$

Step 3.2

Simplify $\pm \sqrt{40}$ .

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Step 3.2.1

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 3.2.1.1

Factor $4$ out of $40$ .

$x = \pm \sqrt{4 (10)}$

Step 3.2.1.2

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2} \cdot 10}$

Step 3.2.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt{10}$

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.

$x = 2 \sqrt{10}$

Step 3.3.2

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

$x = - 2 \sqrt{10}$

Step 3.3.3

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

$x = 2 \sqrt{10}, - 2 \sqrt{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{10}, - 2 \sqrt{10}$

Decimal Form:

$x = 6.32455532 \dots, - 6.32455532 \dots$