Pre-Algebra Examples

$(0.87, - 2.87)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $- 2.87$ of the point, and set $x$ to the $x$ value $0.87$ of the point.

$- 2.87 = a^{0.87}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{0.87} = - 2.87$ .

$a^{0.87} = - 2.87$

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 are $2$ numbers to the right of the decimal point, place the decimal number over $10^{2}$ $(100)$ . Next, add the whole number to the left of the decimal.

$a^{0 \frac{87}{100}} = - 2.87$

Step 2.2.2

Convert $0 \frac{87}{100}$ 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.

$a^{0 + \frac{87}{100}} = - 2.87$

Step 2.2.2.2

Add $0$ and $\frac{87}{100}$ .

$a^{\frac{87}{100}} = - 2.87$

Step 2.3

Raise each side of the equation to the power of $\frac{1}{0.87}$ to eliminate the fractional exponent on the left side.

${(a^{\frac{87}{100}})}^{\frac{1}{0.87}} = {(- 2.87)}^{\frac{1}{0.87}}$

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 ${(a^{\frac{87}{100}})}^{\frac{1}{0.87}}$ .

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Step 2.4.1.1.1

Multiply the exponents in ${(a^{\frac{87}{100}})}^{\frac{1}{0.87}}$ .

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Step 2.4.1.1.1.1

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

$a^{\frac{87}{100} \cdot \frac{1}{0.87}} = {(- 2.87)}^{\frac{1}{0.87}}$

Step 2.4.1.1.1.2

Cancel the common factor of $0.87$ .

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Step 2.4.1.1.1.2.1

Factor $0.87$ out of $87$ .

$a^{\frac{0.87 (100)}{100} \cdot \frac{1}{0.87}} = {(- 2.87)}^{\frac{1}{0.87}}$

Step 2.4.1.1.1.2.2

Cancel the common factor.

$a^{\frac{0.87 \cdot 100}{100} \cdot \frac{1}{0.87}} = {(- 2.87)}^{\frac{1}{0.87}}$

Step 2.4.1.1.1.2.3

Rewrite the expression.

$a^{\frac{100}{100}} = {(- 2.87)}^{\frac{1}{0.87}}$

Step 2.4.1.1.1.3

Divide $100$ by $100$ .

$a^{1} = {(- 2.87)}^{\frac{1}{0.87}}$

Step 2.4.1.1.2

Simplify.

$a = {(- 2.87)}^{\frac{1}{0.87}}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Divide $1$ by $0.87$ .

$a = {(- 2.87)}^{1.14942528}$

Step 2.5

Exclude the solutions that do not make $- 2.87 = a^{0.87}$ true.

$No solution$

Step 3

Since there is no real solution, the exponential function cannot be found.

The exponential function cannot be found