Finite Math Examples

$(- 6.8, 7 (- 2.1 - 6.2))$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $7 (- 2.1 - 6.2)$ of the point, and set $x$ to the $x$ value $- 6.8$ of the point.

$7 (- 2.1 - 6.2) = a^{- 6.8}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{- 6.8} = 7 (- 2.1 - 6.2)$ .

$a^{- 6.8} = 7 (- 2.1 - 6.2)$

Step 2.2

Move the terms containing $a$ to the left side and simplify.

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Step 2.2.1

Subtract $6.2$ from $- 2.1$ .

$a^{- 6.8} = 7 \cdot - 8.3$

Step 2.2.2

Multiply $7$ by $- 8.3$ .

$a^{- 6.8} = - 58.1$

Step 2.3

Convert the decimal exponent to a fractional exponent.

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Step 2.3.1

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.

$a^{- 6 \frac{8}{10}} = - 58.1$

Step 2.3.2

Reduce the fractional part of the mixed number.

$a^{- 6 \frac{4}{5}} = - 58.1$

Step 2.3.3

Convert $6 \frac{4}{5}$ to an improper fraction.

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Step 2.3.3.1

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

$a^{- (6 + \frac{4}{5})} = - 58.1$

Step 2.3.3.2

Add $6$ and $\frac{4}{5}$ .

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Step 2.3.3.2.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a^{- (6 \cdot \frac{5}{5} + \frac{4}{5})} = - 58.1$

Step 2.3.3.2.2

Combine $6$ and $\frac{5}{5}$ .

$a^{- (\frac{6 \cdot 5}{5} + \frac{4}{5})} = - 58.1$

Step 2.3.3.2.3

Combine the numerators over the common denominator.

$a^{- \frac{6 \cdot 5 + 4}{5}} = - 58.1$

Step 2.3.3.2.4

Simplify the numerator.

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Step 2.3.3.2.4.1

Multiply $6$ by $5$ .

$a^{- \frac{30 + 4}{5}} = - 58.1$

Step 2.3.3.2.4.2

Add $30$ and $4$ .

$a^{- \frac{34}{5}} = - 58.1$

Step 2.4

Raise each side of the equation to the power of $\frac{1}{- 6.8}$ to eliminate the fractional exponent on the left side.

${(a^{- \frac{34}{5}})}^{\frac{1}{- 6.8}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5

Simplify the exponent.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Simplify ${(a^{- \frac{34}{5}})}^{\frac{1}{- 6.8}}$ .

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Step 2.5.1.1.1

Multiply the exponents in ${(a^{- \frac{34}{5}})}^{\frac{1}{- 6.8}}$ .

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Step 2.5.1.1.1.1

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

$a^{- \frac{34}{5} \cdot \frac{1}{- 6.8}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.2

Cancel the common factor of $6.8$ .

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Step 2.5.1.1.1.2.1

Move the leading negative in $- \frac{34}{5}$ into the numerator.

$a^{\frac{- 34}{5} \cdot \frac{1}{- 6.8}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.2.2

Factor $6.8$ out of $- 34$ .

$a^{\frac{6.8 (- 5)}{5} \cdot \frac{1}{- 6.8}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.2.3

Factor $6.8$ out of $- 6.8$ .

$a^{\frac{6.8 \cdot - 5}{5} \cdot \frac{1}{6.8 \cdot - 1}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.2.4

Cancel the common factor.

$a^{\frac{6.8 \cdot - 5}{5} \cdot \frac{1}{6.8 \cdot - 1}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.2.5

Rewrite the expression.

$a^{\frac{- 5}{5} \cdot \frac{1}{- 1}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.3

Multiply $\frac{- 5}{5}$ by $\frac{1}{- 1}$ .

$a^{\frac{- 5}{5 \cdot - 1}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.4

Multiply $5$ by $- 1$ .

$a^{\frac{- 5}{- 5}} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.1.5

Divide $- 5$ by $- 5$ .

$a^{1} = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.1.1.2

Simplify.

$a = {(- 58.1)}^{\frac{1}{- 6.8}}$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify ${(- 58.1)}^{\frac{1}{- 6.8}}$ .

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Step 2.5.2.1.1

Divide $1$ by $- 6.8$ .

$a = {(- 58.1)}^{- 0.14705882}$

Step 2.5.2.1.2

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

$a = \frac{1}{{(- 58.1)}^{0.14705882}}$

Step 2.6

Exclude the solutions that do not make $7 (- 2.1 - 6.2) = a^{- 6.8}$ true.

$No solution$

Step 3

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

The exponential function cannot be found