Calculus Examples

$3^{2.9} = 1708.8422 \cdot x^{- 0.6}$

Step 1

Rewrite the equation as $1708.8422 \cdot x^{- 0.6} = 3^{2.9}$ .

$1708.8422 \cdot x^{- 0.6} = 3^{2.9}$

Step 2

Divide each term in $1708.8422 \cdot x^{- 0.6} = 3^{2.9}$ by $1708.8422$ and simplify.

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Step 2.1

Divide each term in $1708.8422 \cdot x^{- 0.6} = 3^{2.9}$ by $1708.8422$ .

$\frac{1708.8422 \cdot x^{- 0.6}}{1708.8422} = \frac{3^{2.9}}{1708.8422}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Move $x^{- 0.6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1708.8422}{1708.8422 x^{0.6}} = \frac{3^{2.9}}{1708.8422}$

Step 2.2.2

Cancel the common factor of $1708.8422$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{1708.8422}{1708.8422 x^{0.6}} = \frac{3^{2.9}}{1708.8422}$

Step 2.2.2.2

Rewrite the expression.

$\frac{1}{x^{0.6}} = \frac{3^{2.9}}{1708.8422}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $3$ to the power of $2.9$ .

$\frac{1}{x^{0.6}} = \frac{24.19087841}{1708.8422}$

Step 2.3.2

Divide $24.19087841$ by $1708.8422$ .

$\frac{1}{x^{0.6}} = 0.01415629$

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.

$x^{0.6}, 1$

Step 3.2

The LCM of one and any expression is the expression.

$x^{0.6}$

Step 4

Multiply each term in $\frac{1}{x^{0.6}} = 0.01415629$ by $x^{0.6}$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{1}{x^{0.6}} = 0.01415629$ by $x^{0.6}$ .

$\frac{1}{x^{0.6}} x^{0.6} = 0.01415629 x^{0.6}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $x^{0.6}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{1}{x^{0.6}} x^{0.6} = 0.01415629 x^{0.6}$

Step 4.2.1.2

Rewrite the expression.

$1 = 0.01415629 x^{0.6}$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $0.01415629 x^{0.6} = 1$ .

$0.01415629 x^{0.6} = 1$

Step 5.2

Divide each term in $0.01415629 x^{0.6} = 1$ by $0.01415629$ and simplify.

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Step 5.2.1

Divide each term in $0.01415629 x^{0.6} = 1$ by $0.01415629$ .

$\frac{0.01415629 x^{0.6}}{0.01415629} = \frac{1}{0.01415629}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $0.01415629$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{0.01415629 x^{0.6}}{0.01415629} = \frac{1}{0.01415629}$

Step 5.2.2.1.2

Divide $x^{0.6}$ by $1$ .

$x^{0.6} = \frac{1}{0.01415629}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $0.01415629$ .

$x^{0.6} = 70.63994$

Step 5.3

Convert the decimal exponent to a fractional exponent.

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Step 5.3.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

$x^{0 \frac{6}{10}} = 70.63994$

Step 5.3.2

Reduce the fraction.

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Step 5.3.2.1

Convert $0 \frac{6}{10}$ to an improper fraction.

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Step 5.3.2.1.1

A mixed number is an addition of its whole and fractional parts.

$x^{0 + \frac{6}{10}} = 70.63994$

Step 5.3.2.1.2

Add $0$ and $\frac{6}{10}$ .

$x^{\frac{6}{10}} = 70.63994$

Step 5.3.2.2

Cancel the common factor of $6$ and $10$ .

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Step 5.3.2.2.1

Factor $2$ out of $6$ .

$x^{\frac{2 (3)}{10}} = 70.63994$

Step 5.3.2.2.2

Cancel the common factors.

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Step 5.3.2.2.2.1

Factor $2$ out of $10$ .

$x^{\frac{2 \cdot 3}{2 \cdot 5}} = 70.63994$

Step 5.3.2.2.2.2

Cancel the common factor.

$x^{\frac{2 \cdot 3}{2 \cdot 5}} = 70.63994$

Step 5.3.2.2.2.3

Rewrite the expression.

$x^{\frac{3}{5}} = 70.63994$

Step 5.4

Raise each side of the equation to the power of $\frac{1}{0.6}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{3}{5}})}^{\frac{1}{0.6}} = {70.63994}^{\frac{1}{0.6}}$

Step 5.5

Simplify the exponent.

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Step 5.5.1

Simplify the left side.

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Step 5.5.1.1

Simplify ${(x^{\frac{3}{5}})}^{\frac{1}{0.6}}$ .

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Step 5.5.1.1.1

Multiply the exponents in ${(x^{\frac{3}{5}})}^{\frac{1}{0.6}}$ .

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Step 5.5.1.1.1.1

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

$x^{\frac{3}{5} \cdot \frac{1}{0.6}} = {70.63994}^{\frac{1}{0.6}}$

Step 5.5.1.1.1.2

Cancel the common factor of $0.6$ .

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Step 5.5.1.1.1.2.1

Factor $0.6$ out of $3$ .

$x^{\frac{0.6 (5)}{5} \cdot \frac{1}{0.6}} = {70.63994}^{\frac{1}{0.6}}$

Step 5.5.1.1.1.2.2

Cancel the common factor.

$x^{\frac{0.6 \cdot 5}{5} \cdot \frac{1}{0.6}} = {70.63994}^{\frac{1}{0.6}}$

Step 5.5.1.1.1.2.3

Rewrite the expression.

$x^{\frac{5}{5}} = {70.63994}^{\frac{1}{0.6}}$

Step 5.5.1.1.1.3

Divide $5$ by $5$ .

$x^{1} = {70.63994}^{\frac{1}{0.6}}$

Step 5.5.1.1.2

Simplify.

$x = {70.63994}^{\frac{1}{0.6}}$

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Simplify ${70.63994}^{\frac{1}{0.6}}$ .

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Step 5.5.2.1.1

Divide $1$ by $0.6$ .

$x = {70.63994}^{1.\overline{6}}$

Step 5.5.2.1.2

Raise $70.63994$ to the power of $1.\overline{6}$ .

$x = 1207.12025207$