Pre-Algebra Examples

$(- 3, - 8)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $- 8$ of the point, and set $x$ to the $x$ value $- 3$ of the point.

$- 8 = a^{- 3}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{- 3} = - 8$ .

$a^{- 3} = - 8$

Step 2.2

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

$\frac{1}{a^{3}} = - 8$

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.

$a^{3}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$a^{3}$

Step 2.4

Multiply each term in $\frac{1}{a^{3}} = - 8$ by $a^{3}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{1}{a^{3}} = - 8$ by $a^{3}$ .

$\frac{1}{a^{3}} a^{3} = - 8 a^{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $a^{3}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{1}{a^{3}} a^{3} = - 8 a^{3}$

Step 2.4.2.1.2

Rewrite the expression.

$1 = - 8 a^{3}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 8 a^{3} = 1$ .

$- 8 a^{3} = 1$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$- 8 a^{3} - 1 = 0$

Step 2.5.3

Factor the left side of the equation.

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Step 2.5.3.1

Factor $- 1$ out of $- 8 a^{3} - 1$ .

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Step 2.5.3.1.1

Factor $- 1$ out of $- 8 a^{3}$ .

$- (8 a^{3}) - 1 = 0$

Step 2.5.3.1.2

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

$- (8 a^{3}) - 1 \cdot 1 = 0$

Step 2.5.3.1.3

Factor $- 1$ out of $- (8 a^{3}) - 1 (1)$ .

$- (8 a^{3} + 1) = 0$

Step 2.5.3.2

Rewrite $8 a^{3}$ as ${(2 a)}^{3}$ .

$- ({(2 a)}^{3} + 1) = 0$

Step 2.5.3.3

Rewrite $1$ as $1^{3}$ .

$- ({(2 a)}^{3} + 1^{3}) = 0$

Step 2.5.3.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 2 a$ and $b = 1$ .

$- ((2 a + 1) ({(2 a)}^{2} - 2 a \cdot 1 + 1^{2})) = 0$

Step 2.5.3.5

Factor.

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Step 2.5.3.5.1

Simplify.

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Step 2.5.3.5.1.1

Apply the product rule to $2 a$ .

$- ((2 a + 1) (2^{2} a^{2} - 2 a \cdot 1 + 1^{2})) = 0$

Step 2.5.3.5.1.2

Raise $2$ to the power of $2$ .

$- ((2 a + 1) (4 a^{2} - 2 a \cdot 1 + 1^{2})) = 0$

Step 2.5.3.5.1.3

Multiply $2$ by $- 1$ .

$- ((2 a + 1) (4 a^{2} - 2 a \cdot 1 + 1^{2})) = 0$

Step 2.5.3.5.1.4

Multiply $- 2$ by $1$ .

$- ((2 a + 1) (4 a^{2} - 2 a + 1^{2})) = 0$

Step 2.5.3.5.1.5

One to any power is one.

$- ((2 a + 1) (4 a^{2} - 2 a + 1)) = 0$

Step 2.5.3.5.2

Remove unnecessary parentheses.

$- (2 a + 1) (4 a^{2} - 2 a + 1) = 0$

Step 2.5.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 a + 1 = 0$

$4 a^{2} - 2 a + 1 = 0$

Step 2.5.5

Set $2 a + 1$ equal to $0$ and solve for $a$ .

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Step 2.5.5.1

Set $2 a + 1$ equal to $0$ .

$2 a + 1 = 0$

Step 2.5.5.2

Solve $2 a + 1 = 0$ for $a$ .

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Step 2.5.5.2.1

Subtract $1$ from both sides of the equation.

$2 a = - 1$

Step 2.5.5.2.2

Divide each term in $2 a = - 1$ by $2$ and simplify.

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Step 2.5.5.2.2.1

Divide each term in $2 a = - 1$ by $2$ .

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

Step 2.5.5.2.2.2

Simplify the left side.

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Step 2.5.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.5.2.2.2.1.1

Cancel the common factor.

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

Step 2.5.5.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 2.5.5.2.2.3

Simplify the right side.

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Step 2.5.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.5.6

Set $4 a^{2} - 2 a + 1$ equal to $0$ and solve for $a$ .

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Step 2.5.6.1

Set $4 a^{2} - 2 a + 1$ equal to $0$ .

$4 a^{2} - 2 a + 1 = 0$

Step 2.5.6.2

Solve $4 a^{2} - 2 a + 1 = 0$ for $a$ .

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Step 2.5.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.6.2.2

Substitute the values $a = 4$ , $b = - 2$ , and $c = 1$ into the quadratic formula and solve for $a$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (4 \cdot 1)}}{2 \cdot 4}$

Step 2.5.6.2.3

Simplify.

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Step 2.5.6.2.3.1

Simplify the numerator.

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Step 2.5.6.2.3.1.1

Raise $- 2$ to the power of $2$ .

$a = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$

Step 2.5.6.2.3.1.2

Multiply $- 4 \cdot 4 \cdot 1$ .

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Step 2.5.6.2.3.1.2.1

Multiply $- 4$ by $4$ .

$a = \frac{2 \pm \sqrt{4 - 16 \cdot 1}}{2 \cdot 4}$

Step 2.5.6.2.3.1.2.2

Multiply $- 16$ by $1$ .

$a = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

Step 2.5.6.2.3.1.3

Subtract $16$ from $4$ .

$a = \frac{2 \pm \sqrt{- 12}}{2 \cdot 4}$

Step 2.5.6.2.3.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$a = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

Step 2.5.6.2.3.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$a = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

Step 2.5.6.2.3.1.6

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

$a = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

Step 2.5.6.2.3.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.5.6.2.3.1.7.1

Factor $4$ out of $12$ .

$a = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 4}$

Step 2.5.6.2.3.1.7.2

Rewrite $4$ as $2^{2}$ .

$a = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

Step 2.5.6.2.3.1.8

Pull terms out from under the radical.

$a = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 4}$

Step 2.5.6.2.3.1.9

Move $2$ to the left of $i$ .

$a = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

Step 2.5.6.2.3.2

Multiply $2$ by $4$ .

$a = \frac{2 \pm 2 i \sqrt{3}}{8}$

Step 2.5.6.2.3.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{8}$ .

$a = \frac{1 \pm i \sqrt{3}}{4}$

Step 2.5.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.6.2.4.1

Simplify the numerator.

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Step 2.5.6.2.4.1.1

Raise $- 2$ to the power of $2$ .

$a = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$

Step 2.5.6.2.4.1.2

Multiply $- 4 \cdot 4 \cdot 1$ .

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Step 2.5.6.2.4.1.2.1

Multiply $- 4$ by $4$ .

$a = \frac{2 \pm \sqrt{4 - 16 \cdot 1}}{2 \cdot 4}$

Step 2.5.6.2.4.1.2.2

Multiply $- 16$ by $1$ .

$a = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

Step 2.5.6.2.4.1.3

Subtract $16$ from $4$ .

$a = \frac{2 \pm \sqrt{- 12}}{2 \cdot 4}$

Step 2.5.6.2.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$a = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

Step 2.5.6.2.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$a = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

Step 2.5.6.2.4.1.6

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

$a = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

Step 2.5.6.2.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.5.6.2.4.1.7.1

Factor $4$ out of $12$ .

$a = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 4}$

Step 2.5.6.2.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$a = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

Step 2.5.6.2.4.1.8

Pull terms out from under the radical.

$a = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 4}$

Step 2.5.6.2.4.1.9

Move $2$ to the left of $i$ .

$a = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

Step 2.5.6.2.4.2

Multiply $2$ by $4$ .

$a = \frac{2 \pm 2 i \sqrt{3}}{8}$

Step 2.5.6.2.4.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{8}$ .

$a = \frac{1 \pm i \sqrt{3}}{4}$

Step 2.5.6.2.4.4

Change the $\pm$ to $+$ .

$a = \frac{1 + i \sqrt{3}}{4}$

Step 2.5.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.6.2.5.1

Simplify the numerator.

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Step 2.5.6.2.5.1.1

Raise $- 2$ to the power of $2$ .

$a = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$

Step 2.5.6.2.5.1.2

Multiply $- 4 \cdot 4 \cdot 1$ .

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Step 2.5.6.2.5.1.2.1

Multiply $- 4$ by $4$ .

$a = \frac{2 \pm \sqrt{4 - 16 \cdot 1}}{2 \cdot 4}$

Step 2.5.6.2.5.1.2.2

Multiply $- 16$ by $1$ .

$a = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

Step 2.5.6.2.5.1.3

Subtract $16$ from $4$ .

$a = \frac{2 \pm \sqrt{- 12}}{2 \cdot 4}$

Step 2.5.6.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$a = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

Step 2.5.6.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$a = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

Step 2.5.6.2.5.1.6

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

$a = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

Step 2.5.6.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.5.6.2.5.1.7.1

Factor $4$ out of $12$ .

$a = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 4}$

Step 2.5.6.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$a = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

Step 2.5.6.2.5.1.8

Pull terms out from under the radical.

$a = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 4}$

Step 2.5.6.2.5.1.9

Move $2$ to the left of $i$ .

$a = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

Step 2.5.6.2.5.2

Multiply $2$ by $4$ .

$a = \frac{2 \pm 2 i \sqrt{3}}{8}$

Step 2.5.6.2.5.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{8}$ .

$a = \frac{1 \pm i \sqrt{3}}{4}$

Step 2.5.6.2.5.4

Change the $\pm$ to $-$ .

$a = \frac{1 - i \sqrt{3}}{4}$

Step 2.5.6.2.6

The final answer is the combination of both solutions.

$a = \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 2.5.7

The final solution is all the values that make $- (2 a + 1) (4 a^{2} - 2 a + 1) = 0$ true.

$a = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 2.6

Remove all values that contain imaginary components.

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Step 2.6.1

There are no imaginary components. Add $- \frac{1}{2}$ to the final answer.

$- \frac{1}{2}$ is a real number

Step 2.6.2

The letter $i$ represents an imaginary component, and is not a real number. Do not add $\frac{1 + i \sqrt{3}}{4}$ to the final answer.

$\frac{1 + i \sqrt{3}}{4}$ is not a real number

Step 2.6.3

The letter $i$ represents an imaginary component, and is not a real number. Do not add $\frac{1 - i \sqrt{3}}{4}$ to the final answer.

$\frac{1 - i \sqrt{3}}{4}$ is not a real number

Step 2.6.4

The final answer is the list of values not containing imaginary components.

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

Step 3

Substitute each value for $a$ back into the function $f (x) = a^{x}$ to find each possible exponential function.

$f (x) = {(- \frac{1}{2})}^{x}$