Basic Math Examples

$15 = \sqrt{j^{2} - 400^{2}}$

Step 1

Rewrite the equation as $\sqrt{j^{2} - 400^{2}} = 15$ .

$\sqrt{j^{2} - 400^{2}} = 15$

Step 2

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

${\sqrt{j^{2} - 400^{2}}}^{2} = 15^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{j^{2} - 400^{2}}$ as ${(j^{2} - 400^{2})}^{\frac{1}{2}}$ .

${({(j^{2} - 400^{2})}^{\frac{1}{2}})}^{2} = 15^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(j^{2} - 400^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(j^{2} - 400^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(j^{2} - 400^{2})}^{\frac{1}{2} \cdot 2} = 15^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(j^{2} - 400^{2})}^{\frac{1}{2} \cdot 2} = 15^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(j^{2} - 400^{2})}^{1} = 15^{2}$

Step 3.2.1.2

Simplify each term.

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Step 3.2.1.2.1

Raise $400$ to the power of $2$ .

${(j^{2} - 1 \cdot 160000)}^{1} = 15^{2}$

Step 3.2.1.2.2

Multiply $- 1$ by $160000$ .

${(j^{2} - 160000)}^{1} = 15^{2}$

Step 3.2.1.3

Simplify.

$j^{2} - 160000 = 15^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $15$ to the power of $2$ .

$j^{2} - 160000 = 225$

Step 4

Solve for $j$ .

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Step 4.1

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

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Step 4.1.1

Add $160000$ to both sides of the equation.

$j^{2} = 225 + 160000$

Step 4.1.2

Add $225$ and $160000$ .

$j^{2} = 160225$

Step 4.2

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

$j = \pm \sqrt{160225}$

Step 4.3

Simplify $\pm \sqrt{160225}$ .

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Step 4.3.1

Rewrite $160225$ as $5^{2} \cdot 6409$ .

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Step 4.3.1.1

Factor $25$ out of $160225$ .

$j = \pm \sqrt{25 (6409)}$

Step 4.3.1.2

Rewrite $25$ as $5^{2}$ .

$j = \pm \sqrt{5^{2} \cdot 6409}$

Step 4.3.2

Pull terms out from under the radical.

$j = \pm 5 \sqrt{6409}$

Step 4.4

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

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Step 4.4.1

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

$j = 5 \sqrt{6409}$

Step 4.4.2

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

$j = - 5 \sqrt{6409}$

Step 4.4.3

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

$j = 5 \sqrt{6409}, - 5 \sqrt{6409}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$j = 5 \sqrt{6409}, - 5 \sqrt{6409}$

Decimal Form:

$j = 400.28115119 \dots, - 400.28115119 \dots$