Basic Math Examples

\sqrt{s^{2}} = \sqrt{1040}

$\sqrt{s^{2}} = \sqrt{1040}$

Step 1

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

{\sqrt{s^{2}}}^{2} = {\sqrt{1040}}^{2}

${\sqrt{s^{2}}}^{2} = {\sqrt{1040}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{s^{2}}

$\sqrt{s^{2}}$ as

s^{\frac{2}{2}}

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

{(s^{\frac{2}{2}})}^{2} = {\sqrt{1040}}^{2}

${(s^{\frac{2}{2}})}^{2} = {\sqrt{1040}}^{2}$

Step 2.2

Divide

2

$2$ by

2

$2$ .

{(s^{1})}^{2} = {\sqrt{1040}}^{2}

${(s^{1})}^{2} = {\sqrt{1040}}^{2}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Multiply the exponents in

{(s^{1})}^{2}

${(s^{1})}^{2}$ .

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Step 2.3.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

s^{1 \cdot 2} = {\sqrt{1040}}^{2}

$s^{1 \cdot 2} = {\sqrt{1040}}^{2}$

Step 2.3.1.2

Multiply

2

$2$ by

1

$1$ .

s^{2} = {\sqrt{1040}}^{2}

$s^{2} = {\sqrt{1040}}^{2}$

s^{2} = {\sqrt{1040}}^{2}

$s^{2} = {\sqrt{1040}}^{2}$

s^{2} = {\sqrt{1040}}^{2}

$s^{2} = {\sqrt{1040}}^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify

{\sqrt{1040}}^{2}

${\sqrt{1040}}^{2}$ .

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Step 2.4.1.1

Rewrite

1040

$1040$ as

4^{2} \cdot 65

$4^{2} \cdot 65$ .

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Step 2.4.1.1.1

Factor

16

$16$ out of

1040

$1040$ .

s^{2} = {\sqrt{16 (65)}}^{2}

$s^{2} = {\sqrt{16 (65)}}^{2}$

Step 2.4.1.1.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

s^{2} = {\sqrt{4^{2} \cdot 65}}^{2}

$s^{2} = {\sqrt{4^{2} \cdot 65}}^{2}$

s^{2} = {\sqrt{4^{2} \cdot 65}}^{2}

$s^{2} = {\sqrt{4^{2} \cdot 65}}^{2}$

Step 2.4.1.2

Pull terms out from under the radical.

s^{2} = {(4 \sqrt{65})}^{2}

$s^{2} = {(4 \sqrt{65})}^{2}$

Step 2.4.1.3

Simplify the expression.

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Step 2.4.1.3.1

Apply the product rule to

4 \sqrt{65}

$4 \sqrt{65}$ .

s^{2} = 4^{2} {\sqrt{65}}^{2}

$s^{2} = 4^{2} {\sqrt{65}}^{2}$

Step 2.4.1.3.2

Raise

4

$4$ to the power of

2

$2$ .

s^{2} = 16 {\sqrt{65}}^{2}

$s^{2} = 16 {\sqrt{65}}^{2}$

s^{2} = 16 {\sqrt{65}}^{2}

$s^{2} = 16 {\sqrt{65}}^{2}$

Step 2.4.1.4

Rewrite

{\sqrt{65}}^{2}

${\sqrt{65}}^{2}$ as

65

$65$ .

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Step 2.4.1.4.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{65}

$\sqrt{65}$ as

65^{\frac{1}{2}}

$65^{\frac{1}{2}}$ .

s^{2} = 16 {(65^{\frac{1}{2}})}^{2}

$s^{2} = 16 {(65^{\frac{1}{2}})}^{2}$

Step 2.4.1.4.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

s^{2} = 16 \cdot 65^{\frac{1}{2} \cdot 2}

$s^{2} = 16 \cdot 65^{\frac{1}{2} \cdot 2}$

Step 2.4.1.4.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

s^{2} = 16 \cdot 65^{\frac{2}{2}}

$s^{2} = 16 \cdot 65^{\frac{2}{2}}$

Step 2.4.1.4.4

Cancel the common factor of

2

$2$ .

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Step 2.4.1.4.4.1

Cancel the common factor.

$s^{2} = 16 \cdot 65^{\frac{2}{2}}$

Step 2.4.1.4.4.2

Rewrite the expression.

$s^{2} = 16 \cdot 65^{1}$

Step 2.4.1.4.5

Evaluate the exponent.

$s^{2} = 16 \cdot 65$

Step 2.4.1.5

Multiply

$16$ by

$65$ .

$s^{2} = 1040$

Step 3

Solve for

$s$ .

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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.

$s = \pm \sqrt{1040}$

Step 3.2

Simplify

$\pm \sqrt{1040}$ .

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Step 3.2.1

Rewrite

$1040$ as

$4^{2} \cdot 65$ .

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Step 3.2.1.1

Factor

$16$ out of

$1040$ .

$s = \pm \sqrt{16 (65)}$

Step 3.2.1.2

Rewrite

$16$ as

$4^{2}$ .

$s = \pm \sqrt{4^{2} \cdot 65}$

Step 3.2.2

Pull terms out from under the radical.

$s = \pm 4 \sqrt{65}$

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.

$s = 4 \sqrt{65}$

Step 3.3.2

Next, use the negative value of the

$\pm$ to find the second solution.

$s = - 4 \sqrt{65}$

Step 3.3.3

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

$s = 4 \sqrt{65}, - 4 \sqrt{65}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$s = 4 \sqrt{65}, - 4 \sqrt{65}$

Decimal Form:

$s = 32.24903099 \dots, - 32.24903099 \dots$