Finite Math Examples

$7^{- 6 - x^{2}} = 7^{8}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$7^{- 6 - x^{2}} = 7^{8}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$- 6 - x^{2} = 8$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Add $6$ to both sides of the equation.

$- x^{2} = 8 + 6$

Step 3.1.2

Add $8$ and $6$ .

$- x^{2} = 14$

Step 3.2

Divide each term in $- x^{2} = 14$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x^{2} = 14$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{14}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{14}{- 1}$

Step 3.2.2.2

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

$x^{2} = \frac{14}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $14$ by $- 1$ .

$x^{2} = - 14$

Step 3.3

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

$x = \pm \sqrt{- 14}$

Step 3.4

Simplify $\pm \sqrt{- 14}$ .

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Step 3.4.1

Rewrite $- 14$ as $- 1 (14)$ .

$x = \pm \sqrt{- 1 (14)}$

Step 3.4.2

Rewrite $\sqrt{- 1 (14)}$ as $\sqrt{- 1} \cdot \sqrt{14}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{14}$

Step 3.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{14}$

Step 3.5

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

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Step 3.5.1

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

$x = i \sqrt{14}$

Step 3.5.2

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

$x = - i \sqrt{14}$

Step 3.5.3

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

$x = i \sqrt{14}, - i \sqrt{14}$