Pre-Algebra Examples

$(0.1, 0.08)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $0.08$ of the point, and set $x$ to the $x$ value $0.1$ of the point.

$0.08 = a^{0.1}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{0.1} = 0.08$ .

$a^{0.1} = 0.08$

Step 2.2

Convert the decimal exponent to a fractional exponent.

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Step 2.2.1

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.

$a^{0 \frac{1}{10}} = 0.08$

Step 2.2.2

Convert $0 \frac{1}{10}$ to an improper fraction.

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Step 2.2.2.1

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

$a^{0 + \frac{1}{10}} = 0.08$

Step 2.2.2.2

Add $0$ and $\frac{1}{10}$ .

$a^{\frac{1}{10}} = 0.08$

Step 2.3

Raise each side of the equation to the power of $\frac{1}{0.1}$ to eliminate the fractional exponent on the left side.

${(a^{\frac{1}{10}})}^{\frac{1}{0.1}} = {0.08}^{\frac{1}{0.1}}$

Step 2.4

Simplify the exponent.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify ${(a^{\frac{1}{10}})}^{\frac{1}{0.1}}$ .

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Step 2.4.1.1.1

Multiply the exponents in ${(a^{\frac{1}{10}})}^{\frac{1}{0.1}}$ .

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Step 2.4.1.1.1.1

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

$a^{\frac{1}{10} \cdot \frac{1}{0.1}} = {0.08}^{\frac{1}{0.1}}$

Step 2.4.1.1.1.2

Cancel the common factor of $0.1$ .

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Step 2.4.1.1.1.2.1

Factor $0.1$ out of $1$ .

$a^{\frac{0.1 (10)}{10} \cdot \frac{1}{0.1}} = {0.08}^{\frac{1}{0.1}}$

Step 2.4.1.1.1.2.2

Cancel the common factor.

$a^{\frac{0.1 \cdot 10}{10} \cdot \frac{1}{0.1}} = {0.08}^{\frac{1}{0.1}}$

Step 2.4.1.1.1.2.3

Rewrite the expression.

$a^{\frac{10}{10}} = {0.08}^{\frac{1}{0.1}}$

Step 2.4.1.1.1.3

Divide $10$ by $10$ .

$a^{1} = {0.08}^{\frac{1}{0.1}}$

Step 2.4.1.1.2

Simplify.

$a = {0.08}^{\frac{1}{0.1}}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify ${0.08}^{\frac{1}{0.1}}$ .

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Step 2.4.2.1.1

Divide $1$ by $0.1$ .

$a = {0.08}^{10}$

Step 2.4.2.1.2

Raise $0.08$ to the power of $10$ .

$a = 0$

Step 3

Substitute each value for $a$ back into the function $f (x) = a^{x}$ to find each possible exponential function.

$f (x) = {(0)}^{x}$