Pre-Algebra Examples

$(6, - 3)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $- 3$ of the point, and set $x$ to the $x$ value $6$ of the point.

$- 3 = a^{6}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{6} = - 3$ .

$a^{6} = - 3$

Step 2.2

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

$a = \pm \sqrt[6]{- 3}$

Step 2.3

Simplify $\pm \sqrt[6]{- 3}$ .

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Step 2.3.1

Rewrite $- 3$ as $i^{6} \cdot 3$ .

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Step 2.3.1.1

Rewrite $- 3$ as $- 1 (3)$ .

$a = \pm \sqrt[6]{- 1 (3)}$

Step 2.3.1.2

Rewrite $- 1$ as $i^{6}$ .

$a = \pm \sqrt[6]{i^{6} \cdot 3}$

Step 2.3.2

Pull terms out from under the radical.

$a = \pm i \sqrt[6]{3}$

Step 2.4

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

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Step 2.4.1

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

$a = i \sqrt[6]{3}$

Step 2.4.2

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

$a = - i \sqrt[6]{3}$

Step 2.4.3

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

$a = i \sqrt[6]{3}, - i \sqrt[6]{3}$

Step 2.5

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