Finite Math Examples

$108 x^{5} = 1.3 a - 34$

Step 1

Divide each term in $108 x^{5} = 1.3 a - 34$ by $108$ and simplify.

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Step 1.1

Divide each term in $108 x^{5} = 1.3 a - 34$ by $108$ .

$\frac{108 x^{5}}{108} = \frac{1.3 a}{108} + \frac{- 34}{108}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $108$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{108 x^{5}}{108} = \frac{1.3 a}{108} + \frac{- 34}{108}$

Step 1.2.1.2

Divide $x^{5}$ by $1$ .

$x^{5} = \frac{1.3 a}{108} + \frac{- 34}{108}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Factor $1.3$ out of $1.3 a$ .

$x^{5} = \frac{1.3 (a)}{108} + \frac{- 34}{108}$

Step 1.3.1.2

Factor $108$ out of $108$ .

$x^{5} = \frac{1.3 (a)}{108 (1)} + \frac{- 34}{108}$

Step 1.3.1.3

Separate fractions.

$x^{5} = \frac{1.3}{108} \cdot \frac{a}{1} + \frac{- 34}{108}$

Step 1.3.1.4

Divide $1.3$ by $108$ .

$x^{5} = 0.012\overline{037} \frac{a}{1} + \frac{- 34}{108}$

Step 1.3.1.5

Divide $a$ by $1$ .

$x^{5} = 0.012\overline{037} a + \frac{- 34}{108}$

Step 1.3.1.6

Cancel the common factor of $- 34$ and $108$ .

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Step 1.3.1.6.1

Factor $2$ out of $- 34$ .

$x^{5} = 0.012\overline{037} a + \frac{2 (- 17)}{108}$

Step 1.3.1.6.2

Cancel the common factors.

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Step 1.3.1.6.2.1

Factor $2$ out of $108$ .

$x^{5} = 0.012\overline{037} a + \frac{2 \cdot - 17}{2 \cdot 54}$

Step 1.3.1.6.2.2

Cancel the common factor.

$x^{5} = 0.012\overline{037} a + \frac{2 \cdot - 17}{2 \cdot 54}$

Step 1.3.1.6.2.3

Rewrite the expression.

$x^{5} = 0.012\overline{037} a + \frac{- 17}{54}$

Step 1.3.1.7

Move the negative in front of the fraction.

$x^{5} = 0.012\overline{037} a - \frac{17}{54}$

Step 2

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

$x = \sqrt[5]{0.012\overline{037} a - \frac{17}{54}}$

Step 3

Simplify $\sqrt[5]{0.012\overline{037} a - \frac{17}{54}}$ .

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Step 3.1

To write $0.012\overline{037} a$ as a fraction with a common denominator, multiply by $\frac{54}{54}$ .

$x = \sqrt[5]{0.012\overline{037} a \cdot \frac{54}{54} - \frac{17}{54}}$

Step 3.2

Combine $0.012\overline{037} a$ and $\frac{54}{54}$ .

$x = \sqrt[5]{\frac{0.012\overline{037} a \cdot 54}{54} - \frac{17}{54}}$

Step 3.3

Combine the numerators over the common denominator.

$x = \sqrt[5]{\frac{0.012\overline{037} a \cdot 54 - 17}{54}}$

Step 3.4

Multiply $54$ by $0.012\overline{037}$ .

$x = \sqrt[5]{\frac{0.64\overline{9} a - 17}{54}}$

Step 3.5

Rewrite $\sqrt[5]{\frac{0.64\overline{9} a - 17}{54}}$ as $\frac{\sqrt[5]{0.64\overline{9} a - 17}}{\sqrt[5]{54}}$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17}}{\sqrt[5]{54}}$

Step 3.6

Multiply $\frac{\sqrt[5]{0.64\overline{9} a - 17}}{\sqrt[5]{54}}$ by $\frac{{\sqrt[5]{54}}^{4}}{{\sqrt[5]{54}}^{4}}$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17}}{\sqrt[5]{54}} \cdot \frac{{\sqrt[5]{54}}^{4}}{{\sqrt[5]{54}}^{4}}$

Step 3.7

Combine and simplify the denominator.

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Step 3.7.1

Multiply $\frac{\sqrt[5]{0.64\overline{9} a - 17}}{\sqrt[5]{54}}$ by $\frac{{\sqrt[5]{54}}^{4}}{{\sqrt[5]{54}}^{4}}$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{\sqrt[5]{54} {\sqrt[5]{54}}^{4}}$

Step 3.7.2

Raise $\sqrt[5]{54}$ to the power of $1$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{{\sqrt[5]{54}}^{1} {\sqrt[5]{54}}^{4}}$

Step 3.7.3

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

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{{\sqrt[5]{54}}^{1 + 4}}$

Step 3.7.4

Add $1$ and $4$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{{\sqrt[5]{54}}^{5}}$

Step 3.7.5

Rewrite ${\sqrt[5]{54}}^{5}$ as $54$ .

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Step 3.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{54}$ as $54^{\frac{1}{5}}$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{{(54^{\frac{1}{5}})}^{5}}$

Step 3.7.5.2

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

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{54^{\frac{1}{5} \cdot 5}}$

Step 3.7.5.3

Combine $\frac{1}{5}$ and $5$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{54^{\frac{5}{5}}}$

Step 3.7.5.4

Cancel the common factor of $5$ .

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Step 3.7.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{54^{\frac{5}{5}}}$

Step 3.7.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{54^{1}}$

Step 3.7.5.5

Evaluate the exponent.

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} {\sqrt[5]{54}}^{4}}{54}$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Rewrite ${\sqrt[5]{54}}^{4}$ as $\sqrt[5]{54^{4}}$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} \sqrt[5]{54^{4}}}{54}$

Step 3.8.2

Raise $54$ to the power of $4$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} \sqrt[5]{8503056}}{54}$

Step 3.8.3

Rewrite $8503056$ as $9^{5} \cdot 144$ .

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Step 3.8.3.1

Factor $59049$ out of $8503056$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} \sqrt[5]{59049 (144)}}{54}$

Step 3.8.3.2

Rewrite $59049$ as $9^{5}$ .

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} \sqrt[5]{9^{5} \cdot 144}}{54}$

Step 3.8.4

Pull terms out from under the radical.

$x = \frac{\sqrt[5]{0.64\overline{9} a - 17} \cdot 9 \sqrt[5]{144}}{54}$

Step 3.8.5

Combine using the product rule for radicals.

$x = \frac{9 \sqrt[5]{(0.64\overline{9} a - 17) \cdot 144}}{54}$

Step 3.9

Reduce the expression by cancelling the common factors.

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Step 3.9.1

Cancel the common factor of $9$ and $54$ .

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Step 3.9.1.1

Factor $9$ out of $9 \sqrt[5]{(0.64\overline{9} a - 17) \cdot 144}$ .

$x = \frac{9 (\sqrt[5]{(0.64\overline{9} a - 17) \cdot 144})}{54}$

Step 3.9.1.2

Cancel the common factors.

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Step 3.9.1.2.1

Factor $9$ out of $54$ .

$x = \frac{9 \sqrt[5]{(0.64\overline{9} a - 17) \cdot 144}}{9 \cdot 6}$

Step 3.9.1.2.2

Cancel the common factor.

$x = \frac{9 \sqrt[5]{(0.64\overline{9} a - 17) \cdot 144}}{9 \cdot 6}$

Step 3.9.1.2.3

Rewrite the expression.

$x = \frac{\sqrt[5]{(0.64\overline{9} a - 17) \cdot 144}}{6}$

Step 3.9.2

Reorder factors in $\frac{\sqrt[5]{(0.64\overline{9} a - 17) \cdot 144}}{6}$ .

$x = \frac{\sqrt[5]{144 (0.64\overline{9} a - 17)}}{6}$