Finite Math Examples

$f = \frac{5}{4 L^{2}} \cdot \sqrt{\frac{m}{x^{2}}}$

Step 1

Rewrite the equation as $\frac{5}{4 L^{2}} \cdot \sqrt{\frac{m}{x^{2}}} = f$ .

$\frac{5}{4 L^{2}} \cdot \sqrt{\frac{m}{x^{2}}} = f$

Step 2

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

${(\frac{5}{4 L^{2}} \cdot \sqrt{\frac{m}{x^{2}}})}^{2} = f^{2}$

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 $\sqrt{\frac{m}{x^{2}}}$ as ${(\frac{m}{x^{2}})}^{\frac{1}{2}}$ .

${(\frac{5}{4 L^{2}} \cdot {(\frac{m}{x^{2}})}^{\frac{1}{2}})}^{2} = f^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(\frac{5}{4 L^{2}} \cdot {(\frac{m}{x^{2}})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Combine fractions.

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Step 3.2.1.1.1

Apply the product rule to $\frac{m}{x^{2}}$ .

${(\frac{5}{4 L^{2}} \cdot \frac{m^{\frac{1}{2}}}{{(x^{2})}^{\frac{1}{2}}})}^{2} = f^{2}$

Step 3.2.1.1.2

Combine.

${(\frac{5 m^{\frac{1}{2}}}{4 L^{2} {(x^{2})}^{\frac{1}{2}}})}^{2} = f^{2}$

Step 3.2.1.2

Simplify the denominator.

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Step 3.2.1.2.1

Multiply the exponents in ${(x^{2})}^{\frac{1}{2}}$ .

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Step 3.2.1.2.1.1

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

${(\frac{5 m^{\frac{1}{2}}}{4 L^{2} x^{2 (\frac{1}{2})}})}^{2} = f^{2}$

Step 3.2.1.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.2.1.2.1

Cancel the common factor.

${(\frac{5 m^{\frac{1}{2}}}{4 L^{2} x^{2 (\frac{1}{2})}})}^{2} = f^{2}$

Step 3.2.1.2.1.2.2

Rewrite the expression.

${(\frac{5 m^{\frac{1}{2}}}{4 L^{2} x^{1}})}^{2} = f^{2}$

Step 3.2.1.2.2

Simplify.

${(\frac{5 m^{\frac{1}{2}}}{4 L^{2} x})}^{2} = f^{2}$

Step 3.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.3.1

Apply the product rule to $\frac{5 m^{\frac{1}{2}}}{4 L^{2} x}$ .

$\frac{{(5 m^{\frac{1}{2}})}^{2}}{{(4 L^{2} x)}^{2}} = f^{2}$

Step 3.2.1.3.2

Apply the product rule to $5 m^{\frac{1}{2}}$ .

$\frac{5^{2} {(m^{\frac{1}{2}})}^{2}}{{(4 L^{2} x)}^{2}} = f^{2}$

Step 3.2.1.3.3

Apply the product rule to $4 L^{2} x$ .

$\frac{5^{2} {(m^{\frac{1}{2}})}^{2}}{{(4 L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.3.4

Apply the product rule to $4 L^{2}$ .

$\frac{5^{2} {(m^{\frac{1}{2}})}^{2}}{4^{2} {(L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

Raise $5$ to the power of $2$ .

$\frac{25 {(m^{\frac{1}{2}})}^{2}}{4^{2} {(L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.4.2

Multiply the exponents in ${(m^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.2.1

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

$\frac{25 m^{\frac{1}{2} \cdot 2}}{4^{2} {(L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.4.2.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.2.1

Cancel the common factor.

$\frac{25 m^{\frac{1}{2} \cdot 2}}{4^{2} {(L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.4.2.2.2

Rewrite the expression.

$\frac{25 m^{1}}{4^{2} {(L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.4.3

Simplify.

$\frac{25 m}{4^{2} {(L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.5

Simplify the denominator.

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Step 3.2.1.5.1

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

$\frac{25 m}{16 {(L^{2})}^{2} x^{2}} = f^{2}$

Step 3.2.1.5.2

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

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Step 3.2.1.5.2.1

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

$\frac{25 m}{16 L^{2 \cdot 2} x^{2}} = f^{2}$

Step 3.2.1.5.2.2

Multiply $2$ by $2$ .

$\frac{25 m}{16 L^{4} x^{2}} = f^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

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

$16 L^{4} x^{2}, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$16 L^{4} x^{2}$

Step 4.2

Multiply each term in $\frac{25 m}{16 L^{4} x^{2}} = f^{2}$ by $16 L^{4} x^{2}$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $\frac{25 m}{16 L^{4} x^{2}} = f^{2}$ by $16 L^{4} x^{2}$ .

$\frac{25 m}{16 L^{4} x^{2}} (16 L^{4} x^{2}) = f^{2} (16 L^{4} x^{2})$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Rewrite using the commutative property of multiplication.

$16 \frac{25 m}{16 L^{4} x^{2}} (L^{4} x^{2}) = f^{2} (16 L^{4} x^{2})$

Step 4.2.2.2

Cancel the common factor of $16$ .

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Step 4.2.2.2.1

Factor $16$ out of $16 L^{4} x^{2}$ .

$16 \frac{25 m}{16 (L^{4} x^{2})} (L^{4} x^{2}) = f^{2} (16 L^{4} x^{2})$

Step 4.2.2.2.2

Cancel the common factor.

$16 \frac{25 m}{16 (L^{4} x^{2})} (L^{4} x^{2}) = f^{2} (16 L^{4} x^{2})$

Step 4.2.2.2.3

Rewrite the expression.

$\frac{25 m}{L^{4} x^{2}} (L^{4} x^{2}) = f^{2} (16 L^{4} x^{2})$

Step 4.2.2.3

Cancel the common factor of $L^{4} x^{2}$ .

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Step 4.2.2.3.1

Cancel the common factor.

$\frac{25 m}{L^{4} x^{2}} (L^{4} x^{2}) = f^{2} (16 L^{4} x^{2})$

Step 4.2.2.3.2

Rewrite the expression.

$25 m = f^{2} (16 L^{4} x^{2})$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Rewrite using the commutative property of multiplication.

$25 m = 16 f^{2} L^{4} x^{2}$

Step 4.3

Solve the equation.

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Step 4.3.1

Rewrite the equation as $16 f^{2} L^{4} x^{2} = 25 m$ .

$16 f^{2} L^{4} x^{2} = 25 m$

Step 4.3.2

Divide each term in $16 f^{2} L^{4} x^{2} = 25 m$ by $16 f^{2} L^{4}$ and simplify.

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Step 4.3.2.1

Divide each term in $16 f^{2} L^{4} x^{2} = 25 m$ by $16 f^{2} L^{4}$ .

$\frac{16 f^{2} L^{4} x^{2}}{16 f^{2} L^{4}} = \frac{25 m}{16 f^{2} L^{4}}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $16$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{16 f^{2} L^{4} x^{2}}{16 f^{2} L^{4}} = \frac{25 m}{16 f^{2} L^{4}}$

Step 4.3.2.2.1.2

Rewrite the expression.

$\frac{f^{2} L^{4} x^{2}}{f^{2} L^{4}} = \frac{25 m}{16 f^{2} L^{4}}$

Step 4.3.2.2.2

Cancel the common factor of $f^{2}$ .

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Step 4.3.2.2.2.1

Cancel the common factor.

$\frac{f^{2} L^{4} x^{2}}{f^{2} L^{4}} = \frac{25 m}{16 f^{2} L^{4}}$

Step 4.3.2.2.2.2

Rewrite the expression.

$\frac{L^{4} x^{2}}{L^{4}} = \frac{25 m}{16 f^{2} L^{4}}$

Step 4.3.2.2.3

Cancel the common factor of $L^{4}$ .

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Step 4.3.2.2.3.1

Cancel the common factor.

$\frac{L^{4} x^{2}}{L^{4}} = \frac{25 m}{16 f^{2} L^{4}}$

Step 4.3.2.2.3.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{25 m}{16 f^{2} L^{4}}$

Step 4.3.3

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

$x = \pm \sqrt{\frac{25 m}{16 f^{2} L^{4}}}$

Step 4.3.4

Simplify $\pm \sqrt{\frac{25 m}{16 f^{2} L^{4}}}$ .

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Step 4.3.4.1

Rewrite $\frac{25 m}{16 f^{2} L^{4}}$ as ${(\frac{5}{4 f L^{2}})}^{2} m$ .

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Step 4.3.4.1.1

Factor the perfect power $5^{2}$ out of $25 m$ .

$x = \pm \sqrt{\frac{5^{2} m}{16 f^{2} L^{4}}}$

Step 4.3.4.1.2

Factor the perfect power ${(4 f L^{2})}^{2}$ out of $16 f^{2} L^{4}$ .

$x = \pm \sqrt{\frac{5^{2} m}{{(4 f L^{2})}^{2} \cdot 1}}$

Step 4.3.4.1.3

Rearrange the fraction $\frac{5^{2} m}{{(4 f L^{2})}^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{5}{4 f L^{2}})}^{2} m}$

Step 4.3.4.2

Pull terms out from under the radical.

$x = \pm \frac{5}{4 f L^{2}} \sqrt{m}$

Step 4.3.4.3

Combine $\frac{5}{4 f L^{2}}$ and $\sqrt{m}$ .

$x = \pm \frac{5 \sqrt{m}}{4 f L^{2}}$

Step 4.3.5

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

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Step 4.3.5.1

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

$x = \frac{5 \sqrt{m}}{4 f L^{2}}$

Step 4.3.5.2

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