Finite Math Examples

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

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

Step 1

To find an exponential function,

f (x) = a^{x}

$f (x) = a^{x}$ , containing the point, set

f (x)

$f (x)$ in the function to the

y

$y$ value

7 (- 2.1 - 6.2)

$7 (- 2.1 - 6.2)$ of the point, and set

x

$x$ to the

x

$x$ value

- 6.8

$- 6.8$ of the point.

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

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

Step 2

Solve the equation for

a

$a$ .

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Step 2.1

Rewrite the equation as

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

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

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

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

Step 2.2

Move the terms containing

a

$a$ to the left side and simplify.

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Step 2.2.1

Subtract

6.2

$6.2$ from

- 2.1

$- 2.1$ .

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

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

Step 2.2.2

Multiply

7

$7$ by

- 8.3

$- 8.3$ .

a^{- 6.8} = - 58.1

$a^{- 6.8} = - 58.1$

a^{- 6.8} = - 58.1

$a^{- 6.8} = - 58.1$

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

$1$ number to the right of the decimal point, place the decimal number over

10^{1}

$10^{1}$

(10)

$(10)$ . Next, add the whole number to the left of the decimal.

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

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

Step 2.3.2

Reduce the fractional part of the mixed number.

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

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

Step 2.3.3

Convert

6 \frac{4}{5}

$6 \frac{4}{5}$ 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

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

Step 2.3.3.2

Add

6

$6$ and

\frac{4}{5}

$\frac{4}{5}$ .

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Step 2.3.3.2.1

To write

6

$6$ as a fraction with a common denominator, multiply by

\frac{5}{5}

$\frac{5}{5}$ .

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

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

Step 2.3.3.2.2

Combine

6

$6$ and

\frac{5}{5}

$\frac{5}{5}$ .

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

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

Step 2.3.3.2.3

Combine the numerators over the common denominator.

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

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

Step 2.3.3.2.4

Simplify the numerator.

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Step 2.3.3.2.4.1

Multiply

6

$6$ by

5

$5$ .

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

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

Step 2.3.3.2.4.2

Add

30

$30$ and

4

$4$ .

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

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

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

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

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

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

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

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

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

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

Step 2.4

Raise each side of the equation to the power of

\frac{1}{- 6.8}

$\frac{1}{- 6.8}$ to eliminate the fractional exponent on the left side.

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

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

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}}

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

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Step 2.5.1.1.1

Multiply the exponents in

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

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

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Step 2.5.1.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.5.1.1.1.2

Cancel the common factor of

6.8

$6.8$ .

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Step 2.5.1.1.1.2.1

Move the leading negative in

- \frac{34}{5}

$- \frac{34}{5}$ into the numerator.

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

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

Step 2.5.1.1.1.2.2

Factor

6.8

$6.8$ out of

- 34

$- 34$ .

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

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

Step 2.5.1.1.1.2.3

Factor

6.8

$6.8$ out of

- 6.8

$- 6.8$ .

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

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

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}}

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

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

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

Step 2.5.1.1.1.3

Multiply

\frac{- 5}{5}

$\frac{- 5}{5}$ by

\frac{1}{- 1}

$\frac{1}{- 1}$ .

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

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

Step 2.5.1.1.1.4

Multiply

5

$5$ by

- 1

$- 1$ .

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

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

Step 2.5.1.1.1.5

Divide

- 5

$- 5$ by

- 5

$- 5$ .

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

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

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

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

Step 2.5.1.1.2

Simplify.

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

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

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

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

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

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

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify

{(- 58.1)}^{\frac{1}{- 6.8}}

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

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Step 2.5.2.1.1

Divide

1

$1$ by

- 6.8

$- 6.8$ .

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

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

Step 2.5.2.1.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

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

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

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

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

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

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

Step 2.6

Exclude the solutions that do not make

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

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

No solution

$No solution$

No solution

$No solution$

Step 3

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

The exponential function cannot be found