Algebra Examples

$C = \sqrt{a^{2} + b^{2}}$

Step 1

Rewrite the equation as $\sqrt{a^{2} + b^{2}} = C$ .

$\sqrt{a^{2} + b^{2}} = C$

Step 2

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

${\sqrt{a^{2} + b^{2}}}^{2} = C^{2}$

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 $\sqrt{a^{2} + b^{2}}$ as ${(a^{2} + b^{2})}^{\frac{1}{2}}$ .

${({(a^{2} + b^{2})}^{\frac{1}{2}})}^{2} = C^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(a^{2} + b^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(a^{2} + b^{2})}^{\frac{1}{2} \cdot 2} = C^{2}$

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.

${(a^{2} + b^{2})}^{\frac{1}{2} \cdot 2} = C^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(a^{2} + b^{2})}^{1} = C^{2}$

Step 3.2.1.2

Simplify.

$a^{2} + b^{2} = C^{2}$

Step 4

Solve for $b$ .

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Step 4.1

Subtract $a^{2}$ from both sides of the equation.

$b^{2} = C^{2} - a^{2}$

Step 4.2

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

$b = \pm \sqrt{C^{2} - a^{2}}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = C$ and $b = a$ .

$b = \pm \sqrt{(C + a) (C - a)}$

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.

$b = \sqrt{(C + a) (C - a)}$

Step 4.4.2

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

$b = - \sqrt{(C + a) (C - a)}$

Step 4.4.3

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

$b = \sqrt{(C + a) (C - a)}$

$b = - \sqrt{(C + a) (C - a)}$

$b = \sqrt{(C + a) (C - a)}$

$b = - \sqrt{(C + a) (C - a)}$

$b = \sqrt{(C + a) (C - a)}$

$b = - \sqrt{(C + a) (C - a)}$