Basic Math Examples

Z = \sqrt{R^{2} + \frac{1}{W L}}

$Z = \sqrt{R^{2} + \frac{1}{W L}}$

Step 1

Rewrite the equation as

\sqrt{R^{2} + \frac{1}{W L}} = Z

$\sqrt{R^{2} + \frac{1}{W L}} = Z$ .

\sqrt{R^{2} + \frac{1}{W L}} = Z

$\sqrt{R^{2} + \frac{1}{W L}} = Z$

Step 2

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

{\sqrt{R^{2} + \frac{1}{W L}}}^{2} = Z^{2}

${\sqrt{R^{2} + \frac{1}{W L}}}^{2} = Z^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use

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

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

\sqrt{R^{2} + \frac{1}{W L}}

$\sqrt{R^{2} + \frac{1}{W L}}$ as

{(R^{2} + \frac{1}{W L})}^{\frac{1}{2}}

${(R^{2} + \frac{1}{W L})}^{\frac{1}{2}}$ .

{({(R^{2} + \frac{1}{W L})}^{\frac{1}{2}})}^{2} = Z^{2}

${({(R^{2} + \frac{1}{W L})}^{\frac{1}{2}})}^{2} = Z^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify

{({(R^{2} + \frac{1}{W L})}^{\frac{1}{2}})}^{2}

${({(R^{2} + \frac{1}{W L})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in

{({(R^{2} + \frac{1}{W L})}^{\frac{1}{2}})}^{2}

${({(R^{2} + \frac{1}{W L})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents,

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

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

{(R^{2} + \frac{1}{W L})}^{\frac{1}{2} \cdot 2} = Z^{2}

${(R^{2} + \frac{1}{W L})}^{\frac{1}{2} \cdot 2} = Z^{2}$

Step 3.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(R^{2} + \frac{1}{W L})}^{\frac{1}{2} \cdot 2} = Z^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(R^{2} + \frac{1}{W L})}^{1} = Z^{2}$

Step 3.2.1.2

Simplify.

$R^{2} + \frac{1}{W L} = Z^{2}$

Step 4

Solve for

$W$ .

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Step 4.1

Subtract

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

$\frac{1}{W L} = Z^{2} - R^{2}$

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$W L, 1, 1$

Step 4.2.2

The LCM of one and any expression is the expression.

$W L$

Step 4.3

Multiply each term in

$\frac{1}{W L} = Z^{2} - R^{2}$ by

$W L$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in

$\frac{1}{W L} = Z^{2} - R^{2}$ by

$W L$ .

$\frac{1}{W L} (W L) = Z^{2} (W L) - R^{2} (W L)$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of

$W L$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{1}{W L} (W L) = Z^{2} (W L) - R^{2} (W L)$

Step 4.3.2.1.2

Rewrite the expression.

$1 = Z^{2} (W L) - R^{2} (W L)$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Remove parentheses.

$1 = Z^{2} W L - R^{2} W L$

Step 4.4

Solve the equation.

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Step 4.4.1

Rewrite the equation as

$Z^{2} W L - R^{2} W L = 1$ .

$Z^{2} W L - R^{2} W L = 1$

Step 4.4.2

Factor

$W L$ out of

$Z^{2} W L - R^{2} W L$ .

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Step 4.4.2.1

Factor

$W L$ out of

$Z^{2} W L$ .

$W L (Z^{2}) - R^{2} W L = 1$

Step 4.4.2.2

Factor

$W L$ out of

$- R^{2} W L$ .

$W L (Z^{2}) + W L (- R^{2}) = 1$

Step 4.4.2.3

Factor

$W L$ out of

$W L (Z^{2}) + W L (- R^{2})$ .

$W L (Z^{2} - R^{2}) = 1$

Step 4.4.3

Factor.

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Step 4.4.3.1

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = Z$ and

$b = R$ .

$W L ((Z + R) (Z - R)) = 1$

Step 4.4.3.2

Remove unnecessary parentheses.

$W L (Z + R) (Z - R) = 1$

Step 4.4.4

Divide each term in

$W L (Z + R) (Z - R) = 1$ by

$L (Z + R) (Z - R)$ and simplify.

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Step 4.4.4.1

Divide each term in

$W L (Z + R) (Z - R) = 1$ by

$L (Z + R) (Z - R)$ .

$\frac{W L (Z + R) (Z - R)}{L (Z + R) (Z - R)} = \frac{1}{L (Z + R) (Z - R)}$

Step 4.4.4.2

Simplify the left side.

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Step 4.4.4.2.1

Cancel the common factor of

$L$ .

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Step 4.4.4.2.1.1

Cancel the common factor.

$\frac{W L (Z + R) (Z - R)}{L (Z + R) (Z - R)} = \frac{1}{L (Z + R) (Z - R)}$

Step 4.4.4.2.1.2

Rewrite the expression.

$\frac{(W (Z + R)) (Z - R)}{(Z + R) (Z - R)} = \frac{1}{L (Z + R) (Z - R)}$

Step 4.4.4.2.2

Cancel the common factor of

$Z + R$ .

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Step 4.4.4.2.2.1

Cancel the common factor.

$\frac{W (Z + R) (Z - R)}{(Z + R) (Z - R)} = \frac{1}{L (Z + R) (Z - R)}$

Step 4.4.4.2.2.2

Rewrite the expression.

$\frac{(W) (Z - R)}{Z - R} = \frac{1}{L (Z + R) (Z - R)}$

Step 4.4.4.2.3

Cancel the common factor of

$Z - R$ .

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Step 4.4.4.2.3.1

Cancel the common factor.

$\frac{W (Z - R)}{Z - R} = \frac{1}{L (Z + R) (Z - R)}$

Step 4.4.4.2.3.2

Divide

$W$ by

$1$ .

$W = \frac{1}{L (Z + R) (Z - R)}$