Finite Math Examples

$5 x^{2 - 7} = 3 x$

Step 1

Simplify $5 x^{2 - 7}$ .

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Step 1.1

Rewrite.

$0 + 0 + 5 x^{2 - 7} = 3 x$

Step 1.2

Simplify by adding zeros.

$5 x^{2 - 7} = 3 x$

Step 1.3

Subtract $7$ from $2$ .

$5 x^{- 5} = 3 x$

Step 1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 \frac{1}{x^{5}} = 3 x$

Step 1.5

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

$\frac{5}{x^{5}} = 3 x$

Step 2

Subtract $3 x$ from both sides of the equation.

$\frac{5}{x^{5}} - 3 x = 0$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$x^{5}, 1, 1$

Step 3.2

The LCM of one and any expression is the expression.

$x^{5}$

Step 4

Multiply each term in $\frac{5}{x^{5}} - 3 x = 0$ by $x^{5}$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{5}{x^{5}} - 3 x = 0$ by $x^{5}$ .

$\frac{5}{x^{5}} x^{5} - 3 x \cdot x^{5} = 0 x^{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $x^{5}$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{5}{x^{5}} x^{5} - 3 x \cdot x^{5} = 0 x^{5}$

Step 4.2.1.1.2

Rewrite the expression.

$5 - 3 x \cdot x^{5} = 0 x^{5}$

Step 4.2.1.2

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 4.2.1.2.1

Move $x^{5}$ .

$5 - 3 (x^{5} x) = 0 x^{5}$

Step 4.2.1.2.2

Multiply $x^{5}$ by $x$ .

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Step 4.2.1.2.2.1

Raise $x$ to the power of $1$ .

$5 - 3 (x^{5} x^{1}) = 0 x^{5}$

Step 4.2.1.2.2.2

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

$5 - 3 x^{5 + 1} = 0 x^{5}$

Step 4.2.1.2.3

Add $5$ and $1$ .

$5 - 3 x^{6} = 0 x^{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $0$ by $x^{5}$ .

$5 - 3 x^{6} = 0$

Step 5

Solve the equation.

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Step 5.1

Subtract $5$ from both sides of the equation.

$- 3 x^{6} = - 5$

Step 5.2

Divide each term in $- 3 x^{6} = - 5$ by $- 3$ and simplify.

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Step 5.2.1

Divide each term in $- 3 x^{6} = - 5$ by $- 3$ .

$\frac{- 3 x^{6}}{- 3} = \frac{- 5}{- 3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 3 x^{6}}{- 3} = \frac{- 5}{- 3}$

Step 5.2.2.1.2

Divide $x^{6}$ by $1$ .

$x^{6} = \frac{- 5}{- 3}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Dividing two negative values results in a positive value.

$x^{6} = \frac{5}{3}$

Step 5.3

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

$x = \pm \sqrt[6]{\frac{5}{3}}$

Step 5.4

Rewrite $\sqrt[6]{\frac{5}{3}}$ as $\frac{\sqrt[6]{5}}{\sqrt[6]{3}}$ .

$x = \pm \frac{\sqrt[6]{5}}{\sqrt[6]{3}}$

Step 5.5

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

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Step 5.5.1

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

$x = \frac{\sqrt[6]{5}}{\sqrt[6]{3}}$

Step 5.5.2

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

$x = - \frac{\sqrt[6]{5}}{\sqrt[6]{3}}$

Step 5.5.3

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

$x = \frac{\sqrt[6]{5}}{\sqrt[6]{3}}, - \frac{\sqrt[6]{5}}{\sqrt[6]{3}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[6]{5}}{\sqrt[6]{3}}, - \frac{\sqrt[6]{5}}{\sqrt[6]{3}}$

Decimal Form:

$x = 1.08886688 \dots, - 1.08886688 \dots$