Basic Math Examples

$A = 12 a + 54 \cdot (2 f o i s^{52} f)$

Step 1

Rewrite the equation as $12 a + 54 \cdot (2 f o i s^{52} f) = A$ .

$12 a + 54 \cdot (2 f o i s^{52} f) = A$

Step 2

Simplify each term.

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Step 2.1

Multiply $f$ by $f$ by adding the exponents.

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Step 2.1.1

Move $f$ .

$12 a + 54 \cdot (2 (f \cdot f) o i s^{52}) = A$

Step 2.1.2

Multiply $f$ by $f$ .

$12 a + 54 \cdot (2 f^{2} o i s^{52}) = A$

Step 2.2

Multiply $2$ by $54$ .

$12 a + 108 f^{2} o i s^{52} = A$

Step 3

Subtract $12 a$ from both sides of the equation.

$108 f^{2} o i s^{52} = A - 12 a$

Step 4

Divide each term in $108 f^{2} o i s^{52} = A - 12 a$ by $108 f^{2} o i$ and simplify.

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Step 4.1

Divide each term in $108 f^{2} o i s^{52} = A - 12 a$ by $108 f^{2} o i$ .

$\frac{108 f^{2} o i s^{52}}{108 f^{2} o i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $108$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{108 f^{2} o i s^{52}}{108 f^{2} o i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2.1.2

Rewrite the expression.

$\frac{f^{2} o i s^{52}}{f^{2} o i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2.2

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

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Step 4.2.2.1

Cancel the common factor.

$\frac{f^{2} o i s^{52}}{f^{2} o i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2.2.2

Rewrite the expression.

$\frac{o i s^{52}}{o i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2.3

Cancel the common factor of $o$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{o i s^{52}}{o i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2.3.2

Rewrite the expression.

$\frac{i s^{52}}{i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2.4

Cancel the common factor of $i$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{i s^{52}}{i} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.2.4.2

Divide $s^{52}$ by $1$ .

$s^{52} = \frac{A}{108 f^{2} o i} + \frac{- 12 a}{108 f^{2} o i}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $- 12$ and $108$ .

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Step 4.3.1.1.1

Factor $12$ out of $- 12 a$ .

$s^{52} = \frac{A}{108 f^{2} o i} + \frac{12 (- a)}{108 f^{2} o i}$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

Factor $12$ out of $108 f^{2} o i$ .

$s^{52} = \frac{A}{108 f^{2} o i} + \frac{12 (- a)}{12 (9 f^{2} o i)}$

Step 4.3.1.1.2.2

Cancel the common factor.

$s^{52} = \frac{A}{108 f^{2} o i} + \frac{12 (- a)}{12 (9 f^{2} o i)}$

Step 4.3.1.1.2.3

Rewrite the expression.

$s^{52} = \frac{A}{108 f^{2} o i} + \frac{- a}{9 f^{2} o i}$

Step 4.3.1.2

Move the negative in front of the fraction.

$s^{52} = \frac{A}{108 f^{2} o i} - \frac{a}{9 f^{2} o i}$

Step 5

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

$s = \pm \sqrt[52]{\frac{A}{108 f^{2} o i} - \frac{a}{9 f^{2} o i}}$

Step 6

Simplify $\pm \sqrt[52]{\frac{A}{108 f^{2} o i} - \frac{a}{9 f^{2} o i}}$ .

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Step 6.1

To write $- \frac{a}{9 f^{2} o i}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$s = \pm \sqrt[52]{\frac{A}{108 f^{2} o i} - \frac{a}{9 f^{2} o i} \cdot \frac{12}{12}}$

Step 6.2

Write each expression with a common denominator of $108 o i f^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.1

Multiply $\frac{a}{9 f^{2} o i}$ by $\frac{12}{12}$ .

$s = \pm \sqrt[52]{\frac{A}{108 f^{2} o i} - \frac{a \cdot 12}{9 f^{2} o i \cdot 12}}$

Step 6.2.2

Multiply $12$ by $9$ .

$s = \pm \sqrt[52]{\frac{A}{108 f^{2} o i} - \frac{a \cdot 12}{108 f^{2} o i}}$

Step 6.3

Combine the numerators over the common denominator.

$s = \pm \sqrt[52]{\frac{A - a \cdot 12}{108 f^{2} o i}}$

Step 6.4

Multiply $12$ by $- 1$ .

$s = \pm \sqrt[52]{\frac{A - 12 a}{108 f^{2} o i}}$

Step 6.5

Rewrite $\sqrt[52]{\frac{A - 12 a}{108 f^{2} o i}}$ as $\frac{\sqrt[52]{A - 12 a}}{\sqrt[52]{108 f^{2} o i}}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a}}{\sqrt[52]{108 f^{2} o i}}$

Step 6.6

Multiply $\frac{\sqrt[52]{A - 12 a}}{\sqrt[52]{108 f^{2} o i}}$ by $\frac{{\sqrt[52]{108 f^{2} o i}}^{51}}{{\sqrt[52]{108 f^{2} o i}}^{51}}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a}}{\sqrt[52]{108 f^{2} o i}} \cdot \frac{{\sqrt[52]{108 f^{2} o i}}^{51}}{{\sqrt[52]{108 f^{2} o i}}^{51}}$

Step 6.7

Combine and simplify the denominator.

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Step 6.7.1

Multiply $\frac{\sqrt[52]{A - 12 a}}{\sqrt[52]{108 f^{2} o i}}$ by $\frac{{\sqrt[52]{108 f^{2} o i}}^{51}}{{\sqrt[52]{108 f^{2} o i}}^{51}}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{\sqrt[52]{108 f^{2} o i} {\sqrt[52]{108 f^{2} o i}}^{51}}$

Step 6.7.2

Raise $\sqrt[52]{108 f^{2} o i}$ to the power of $1$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{\sqrt[52]{108 f^{2} o i}}^{1} {\sqrt[52]{108 f^{2} o i}}^{51}}$

Step 6.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{\sqrt[52]{108 f^{2} o i}}^{1 + 51}}$

Step 6.7.4

Add $1$ and $51$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{\sqrt[52]{108 f^{2} o i}}^{52}}$

Step 6.7.5

Rewrite ${\sqrt[52]{108 f^{2} o i}}^{52}$ as $108 f^{2} o i$ .

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Step 6.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[52]{108 f^{2} o i}$ as ${(108 f^{2} o i)}^{\frac{1}{52}}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{({(108 f^{2} o i)}^{\frac{1}{52}})}^{52}}$

Step 6.7.5.2

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

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{(108 f^{2} o i)}^{\frac{1}{52} \cdot 52}}$

Step 6.7.5.3

Combine $\frac{1}{52}$ and $52$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{(108 f^{2} o i)}^{\frac{52}{52}}}$

Step 6.7.5.4

Cancel the common factor of $52$ .

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Step 6.7.5.4.1

Cancel the common factor.

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{(108 f^{2} o i)}^{\frac{52}{52}}}$

Step 6.7.5.4.2

Rewrite the expression.

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{{(108 f^{2} o i)}^{1}}$

Step 6.7.5.5

Simplify.

$s = \pm \frac{\sqrt[52]{A - 12 a} {\sqrt[52]{108 f^{2} o i}}^{51}}{108 f^{2} o i}$

Step 6.8

Simplify the numerator.

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Step 6.8.1

Rewrite ${\sqrt[52]{108 f^{2} o i}}^{51}$ as $\sqrt[52]{{(108 f^{2} o i)}^{51}}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{{(108 f^{2} o i)}^{51}}}{108 f^{2} o i}$

Step 6.8.2

Apply the product rule to $108 f^{2} o i$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{{(108 f^{2} o)}^{51} i^{51}}}{108 f^{2} o i}$

Step 6.8.3

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

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Step 6.8.3.1

Apply the product rule to $108 f^{2} o$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{{(108 f^{2})}^{51} o^{51} i^{51}}}{108 f^{2} o i}$

Step 6.8.3.2

Apply the product rule to $108 f^{2}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} {(f^{2})}^{51} o^{51} i^{51}}}{108 f^{2} o i}$

Step 6.8.4

Multiply the exponents in ${(f^{2})}^{51}$ .

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Step 6.8.4.1

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

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{2 \cdot 51} o^{51} i^{51}}}{108 f^{2} o i}$

Step 6.8.4.2

Multiply $2$ by $51$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} i^{51}}}{108 f^{2} o i}$

Step 6.8.5

Rewrite $i^{51}$ as ${(i^{4})}^{12} (i^{2} \cdot i)$ .

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Step 6.8.5.1

Factor out $i^{48}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} (i^{48} i^{3})}}{108 f^{2} o i}$

Step 6.8.5.2

Rewrite $i^{48}$ as ${(i^{4})}^{12}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} ({(i^{4})}^{12} i^{3})}}{108 f^{2} o i}$

Step 6.8.5.3

Factor out $i^{2}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} ({(i^{4})}^{12} (i^{2} \cdot i))}}{108 f^{2} o i}$

Step 6.8.6

Rewrite $i^{4}$ as $1$ .

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Step 6.8.6.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} ({({(i^{2})}^{2})}^{12} (i^{2} \cdot i))}}{108 f^{2} o i}$

Step 6.8.6.2

Rewrite $i^{2}$ as $- 1$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} ({({(- 1)}^{2})}^{12} (i^{2} \cdot i))}}{108 f^{2} o i}$

Step 6.8.6.3

Raise $- 1$ to the power of $2$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} (1^{12} (i^{2} \cdot i))}}{108 f^{2} o i}$

Step 6.8.7

One to any power is one.

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} (1 (i^{2} \cdot i))}}{108 f^{2} o i}$

Step 6.8.8

Multiply $i^{2} \cdot i$ by $1$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} (i^{2} \cdot i)}}{108 f^{2} o i}$

Step 6.8.9

Rewrite $i^{2}$ as $- 1$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} (- 1 \cdot i)}}{108 f^{2} o i}$

Step 6.8.10

Rewrite $- 1 i$ as $- i$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{108^{51} f^{102} o^{51} \cdot - 1 i}}{108 f^{2} o i}$

Step 6.8.11

Factor out negative.

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{- 108^{51} f^{102} o^{51} i}}{108 f^{2} o i}$

Step 6.8.12

Rewrite $- 108^{51} f^{102} o^{51} i$ as $f^{52} \cdot (- 1 \cdot 108^{51} f^{50} o^{51} i)$ .

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Step 6.8.12.1

Factor out $f^{52}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{- 108^{51} (f^{52} f^{50}) o^{51} i}}{108 f^{2} o i}$

Step 6.8.12.2

Move $108^{51}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{- 1 f^{52} \cdot 108^{51} f^{50} o^{51} i}}{108 f^{2} o i}$

Step 6.8.12.3

Reorder $- 1$ and $f^{52}$ .

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{f^{52} \cdot - 1 \cdot 108^{51} f^{50} o^{51} i}}{108 f^{2} o i}$

Step 6.8.12.4

Add parentheses.

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{f^{52} \cdot - 1 \cdot 108^{51} f^{50} (o^{51} i)}}{108 f^{2} o i}$

Step 6.8.12.5

Add parentheses.

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{f^{52} \cdot - 1 \cdot 108^{51} (f^{50} (o^{51} i))}}{108 f^{2} o i}$

Step 6.8.12.6

Add parentheses.

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{f^{52} \cdot - 1 \cdot (108^{51} (f^{50} (o^{51} i)))}}{108 f^{2} o i}$

Step 6.8.12.7

Add parentheses.

$s = \pm \frac{\sqrt[52]{A - 12 a} \sqrt[52]{f^{52} \cdot (- 1 \cdot 108^{51} f^{50} o^{51} i)}}{108 f^{2} o i}$

Step 6.8.13

Pull terms out from under the radical.

$s = \pm \frac{\sqrt[52]{A - 12 a} f \sqrt[52]{- 1 \cdot 108^{51} f^{50} o^{51} i}}{108 f^{2} o i}$

Step 6.8.14

Combine using the product rule for radicals.

$s = \pm \frac{f \sqrt[52]{(A - 12 a) \cdot - 1 \cdot 108^{51} f^{50} o^{51} i}}{108 f^{2} o i}$

Step 6.9

Reduce the expression by cancelling the common factors.

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Step 6.9.1

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

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Step 6.9.1.1

Factor $f$ out of $f \sqrt[52]{(A - 12 a) \cdot - 1 \cdot 108^{51} f^{50} o^{51} i}$ .

$s = \pm \frac{f (\sqrt[52]{(A - 12 a) \cdot - 1 \cdot 108^{51} f^{50} o^{51} i})}{108 f^{2} o i}$

Step 6.9.1.2

Cancel the common factors.

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Step 6.9.1.2.1

Factor $f$ out of $108 f^{2} o i$ .

$s = \pm \frac{f (\sqrt[52]{(A - 12 a) \cdot - 1 \cdot 108^{51} f^{50} o^{51} i})}{f (108 f o i)}$

Step 6.9.1.2.2

Cancel the common factor.

$s = \pm \frac{f \sqrt[52]{(A - 12 a) \cdot - 1 \cdot 108^{51} f^{50} o^{51} i}}{f (108 f o i)}$

Step 6.9.1.2.3

Rewrite the expression.

$s = \pm \frac{\sqrt[52]{(A - 12 a) \cdot - 1 \cdot 108^{51} f^{50} o^{51} i}}{108 f o i}$

Step 6.9.2

Reorder factors in $\pm \frac{\sqrt[52]{(A - 12 a) \cdot - 1 \cdot 108^{51} f^{50} o^{51} i}}{108 f o i}$ .

$s = \pm \frac{\sqrt[52]{- 108^{51} f^{50} o^{51} i (A - 12 a)}}{108 f o i}$

Step 7

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

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Step 7.1

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

$s = \frac{\sqrt[52]{- 108^{51} f^{50} o^{51} i (A - 12 a)}}{108 f o i}$

Step 7.2

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

$s = - \frac{\sqrt[52]{- 108^{51} f^{50} o^{51} i (A - 12 a)}}{108 f o i}$

Step 7.3

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

$s = \frac{\sqrt[52]{- 108^{51} f^{50} o^{51} i (A - 12 a)}}{108 f o i}$

$s = - \frac{\sqrt[52]{- 108^{51} f^{50} o^{51} i (A - 12 a)}}{108 f o i}$

$s = \frac{\sqrt[52]{- 108^{51} f^{50} o^{51} i (A - 12 a)}}{108 f o i}$

$s = - \frac{\sqrt[52]{- 108^{51} f^{50} o^{51} i (A - 12 a)}}{108 f o i}$