Finite Math Examples

$195 = \sqrt{100^{2} + h^{2}}$

Step 1

Rewrite the equation as $\sqrt{100^{2} + h^{2}} = 195$ .

$\sqrt{100^{2} + h^{2}} = 195$

Step 2

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

${\sqrt{100^{2} + h^{2}}}^{2} = 195^{2}$

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 $\sqrt{100^{2} + h^{2}}$ as ${(100^{2} + h^{2})}^{\frac{1}{2}}$ .

${({(100^{2} + h^{2})}^{\frac{1}{2}})}^{2} = 195^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(100^{2} + h^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(100^{2} + h^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(100^{2} + h^{2})}^{\frac{1}{2} \cdot 2} = 195^{2}$

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.

${(100^{2} + h^{2})}^{\frac{1}{2} \cdot 2} = 195^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(100^{2} + h^{2})}^{1} = 195^{2}$

Step 3.2.1.2

Raise $100$ to the power of $2$ .

${(10000 + h^{2})}^{1} = 195^{2}$

Step 3.2.1.3

Simplify.

$10000 + h^{2} = 195^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $195$ to the power of $2$ .

$10000 + h^{2} = 38025$

Step 4

Solve for $h$ .

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Step 4.1

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

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Step 4.1.1

Subtract $10000$ from both sides of the equation.

$h^{2} = 38025 - 10000$

Step 4.1.2

Subtract $10000$ from $38025$ .

$h^{2} = 28025$

Step 4.2

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

$h = \pm \sqrt{28025}$

Step 4.3

Simplify $\pm \sqrt{28025}$ .

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Step 4.3.1

Rewrite $28025$ as $5^{2} \cdot 1121$ .

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Step 4.3.1.1

Factor $25$ out of $28025$ .

$h = \pm \sqrt{25 (1121)}$

Step 4.3.1.2

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

$h = \pm \sqrt{5^{2} \cdot 1121}$

Step 4.3.2

Pull terms out from under the radical.

$h = \pm 5 \sqrt{1121}$

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.

$h = 5 \sqrt{1121}$

Step 4.4.2

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

$h = - 5 \sqrt{1121}$

Step 4.4.3

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

$h = 5 \sqrt{1121}, - 5 \sqrt{1121}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$h = 5 \sqrt{1121}, - 5 \sqrt{1121}$

Decimal Form:

$h = 167.40669042 \dots, - 167.40669042 \dots$