Basic Math Examples

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

Step 1

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

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

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

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

Step 2.2

Divide $2$ by $2$ .

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

Step 2.3

Simplify the left side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $2$ by $1$ .

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

Step 2.4

Simplify the right side.

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Step 2.4.1

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

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Step 2.4.1.1

Rewrite $1040$ as $4^{2} \cdot 65$ .

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Step 2.4.1.1.1

Factor $16$ out of $1040$ .

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

Step 2.4.1.1.2

Rewrite $16$ as $4^{2}$ .

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

Step 2.4.1.2

Pull terms out from under the radical.

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

Step 2.4.1.3

Simplify the expression.

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Step 2.4.1.3.1

Apply the product rule to $4 \sqrt{65}$ .

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

Step 2.4.1.3.2

Raise $4$ to the power of $2$ .

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

Step 2.4.1.4

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

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Step 2.4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

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

Step 2.4.1.4.2

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

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

Step 2.4.1.4.3

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

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

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.

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

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:

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