Finite Math Examples

$(- 2, - 7)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $- 7$ of the point, and set $x$ to the $x$ value $- 2$ of the point.

$- 7 = a^{- 2}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{- 2} = - 7$ .

$a^{- 2} = - 7$

Step 2.2

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

$\frac{1}{a^{2}} = - 7$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a^{2}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$a^{2}$

Step 2.4

Multiply each term in $\frac{1}{a^{2}} = - 7$ by $a^{2}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{1}{a^{2}} = - 7$ by $a^{2}$ .

$\frac{1}{a^{2}} a^{2} = - 7 a^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $a^{2}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{1}{a^{2}} a^{2} = - 7 a^{2}$

Step 2.4.2.1.2

Rewrite the expression.

$1 = - 7 a^{2}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 7 a^{2} = 1$ .

$- 7 a^{2} = 1$

Step 2.5.2

Divide each term in $- 7 a^{2} = 1$ by $- 7$ and simplify.

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Step 2.5.2.1

Divide each term in $- 7 a^{2} = 1$ by $- 7$ .

$\frac{- 7 a^{2}}{- 7} = \frac{1}{- 7}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $- 7$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 7 a^{2}}{- 7} = \frac{1}{- 7}$

Step 2.5.2.2.1.2

Divide $a^{2}$ by $1$ .

$a^{2} = \frac{1}{- 7}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Move the negative in front of the fraction.

$a^{2} = - \frac{1}{7}$

Step 2.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{- \frac{1}{7}}$

Step 2.5.4

Simplify $\pm \sqrt{- \frac{1}{7}}$ .

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Step 2.5.4.1

Rewrite $- \frac{1}{7}$ as $i^{2} \frac{1^{2}}{7}$ .

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Step 2.5.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$a = \pm \sqrt{i^{2} \frac{1}{7}}$

Step 2.5.4.1.2

Rewrite $1$ as $1^{2}$ .

$a = \pm \sqrt{i^{2} \frac{1^{2}}{7}}$

Step 2.5.4.2

Pull terms out from under the radical.

$a = \pm i \sqrt{\frac{1^{2}}{7}}$

Step 2.5.4.3

One to any power is one.

$a = \pm i \sqrt{\frac{1}{7}}$

Step 2.5.4.4

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$a = \pm i \frac{\sqrt{1}}{\sqrt{7}}$

Step 2.5.4.5

Any root of $1$ is $1$ .

$a = \pm i \frac{1}{\sqrt{7}}$

Step 2.5.4.6

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$a = \pm i (\frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 2.5.4.7

Combine and simplify the denominator.

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Step 2.5.4.7.1

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$a = \pm i \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.5.4.7.2

Raise $\sqrt{7}$ to the power of $1$ .

$a = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 2.5.4.7.3

Raise $\sqrt{7}$ to the power of $1$ .

$a = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 2.5.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.5.4.7.5

Add $1$ and $1$ .

$a = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.5.4.7.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.5.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$a = \pm i \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.5.4.7.6.2

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

$a = \pm i \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.5.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$a = \pm i \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.5.4.7.6.4

Cancel the common factor of $2$ .

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Step 2.5.4.7.6.4.1

Cancel the common factor.

$a = \pm i \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.5.4.7.6.4.2

Rewrite the expression.

$a = \pm i \frac{\sqrt{7}}{7^{1}}$

Step 2.5.4.7.6.5

Evaluate the exponent.

$a = \pm i \frac{\sqrt{7}}{7}$

Step 2.5.4.8

Combine $i$ and $\frac{\sqrt{7}}{7}$ .

$a = \pm \frac{i \sqrt{7}}{7}$

Step 2.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$a = \frac{i \sqrt{7}}{7}$

Step 2.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$a = - \frac{i \sqrt{7}}{7}$

Step 2.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$a = \frac{i \sqrt{7}}{7}, - \frac{i \sqrt{7}}{7}$

Step 2.6

The final answer is the list of values not containing imaginary components. Since all of the solutions are imaginary, there is no real solution.

No solution

Step 3

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

The exponential function cannot be found