Precalculus Examples

$x^{1.2} = 7$

Step 1

Convert the decimal exponent to a fractional exponent.

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Step 1.1

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.

$x^{1 \frac{2}{10}} = 7$

Step 1.2

Reduce the fractional part of the mixed number.

$x^{1 \frac{1}{5}} = 7$

Step 1.3

Convert $1 \frac{1}{5}$ to an improper fraction.

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Step 1.3.1

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

$x^{1 + \frac{1}{5}} = 7$

Step 1.3.2

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

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Step 1.3.2.1

Write $1$ as a fraction with a common denominator.

$x^{\frac{5}{5} + \frac{1}{5}} = 7$

Step 1.3.2.2

Combine the numerators over the common denominator.

$x^{\frac{5 + 1}{5}} = 7$

Step 1.3.2.3

Add $5$ and $1$ .

$x^{\frac{6}{5}} = 7$

Step 2

Raise each side of the equation to the power of $\frac{1}{1.2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{6}{5}})}^{\frac{1}{1.2}} = 7^{\frac{1}{1.2}}$

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify ${(x^{\frac{6}{5}})}^{\frac{1}{1.2}}$ .

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Step 3.1.1.1

Multiply the exponents in ${(x^{\frac{6}{5}})}^{\frac{1}{1.2}}$ .

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Step 3.1.1.1.1

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

$x^{\frac{6}{5} \cdot \frac{1}{1.2}} = 7^{\frac{1}{1.2}}$

Step 3.1.1.1.2

Cancel the common factor of $1.2$ .

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Step 3.1.1.1.2.1

Factor $1.2$ out of $6$ .

$x^{\frac{1.2 (5)}{5} \cdot \frac{1}{1.2}} = 7^{\frac{1}{1.2}}$

Step 3.1.1.1.2.2

Cancel the common factor.

$x^{\frac{1.2 \cdot 5}{5} \cdot \frac{1}{1.2}} = 7^{\frac{1}{1.2}}$

Step 3.1.1.1.2.3

Rewrite the expression.

$x^{\frac{5}{5}} = 7^{\frac{1}{1.2}}$

Step 3.1.1.1.3

Divide $5$ by $5$ .

$x^{1} = 7^{\frac{1}{1.2}}$

Step 3.1.1.2

Simplify.

$x = 7^{\frac{1}{1.2}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $7^{\frac{1}{1.2}}$ .

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Step 3.2.1.1

Divide $1$ by $1.2$ .

$x = 7^{0.8\overline{3}}$

Step 3.2.1.2

Raise $7$ to the power of $0.8\overline{3}$ .

$x = 5.06114018$