Algebra Examples

$(3, \frac{1}{64})$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $\frac{1}{64}$ of the point, and set $x$ to the $x$ value $3$ of the point.

$\frac{1}{64} = a^{3}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{3} = \frac{1}{64}$ .

$a^{3} = \frac{1}{64}$

Step 2.2

Subtract $\frac{1}{64}$ from both sides of the equation.

$a^{3} - \frac{1}{64} = 0$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Rewrite $\frac{1}{64}$ as ${(\frac{1}{4})}^{3}$ .

$a^{3} - {(\frac{1}{4})}^{3} = 0$

Step 2.3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = a$ and $b = \frac{1}{4}$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Combine $a$ and $\frac{1}{4}$ .

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

Step 2.3.3.2

Apply the product rule to $\frac{1}{4}$ .

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

Step 2.3.3.3

One to any power is one.

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

Step 2.3.3.4

Raise $4$ to the power of $2$ .

$(a - \frac{1}{4}) (a^{2} + \frac{a}{4} + \frac{1}{16}) = 0$

Step 2.4

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

$a - \frac{1}{4} = 0$

$a^{2} + \frac{a}{4} + \frac{1}{16} = 0$

Step 2.5

Set $a - \frac{1}{4}$ equal to $0$ and solve for $a$ .

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Step 2.5.1

Set $a - \frac{1}{4}$ equal to $0$ .

$a - \frac{1}{4} = 0$

Step 2.5.2

Add $\frac{1}{4}$ to both sides of the equation.

$a = \frac{1}{4}$

Step 2.6

Set $a^{2} + \frac{a}{4} + \frac{1}{16}$ equal to $0$ and solve for $a$ .

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Step 2.6.1

Set $a^{2} + \frac{a}{4} + \frac{1}{16}$ equal to $0$ .

$a^{2} + \frac{a}{4} + \frac{1}{16} = 0$

Step 2.6.2

Solve $a^{2} + \frac{a}{4} + \frac{1}{16} = 0$ for $a$ .

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Step 2.6.2.1

Multiply through by the least common denominator $16$ , then simplify.

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Step 2.6.2.1.1

Apply the distributive property.

$16 a^{2} + 16 (\frac{a}{4}) + 16 (\frac{1}{16}) = 0$

Step 2.6.2.1.2

Simplify.

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Step 2.6.2.1.2.1

Cancel the common factor of $4$ .

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Step 2.6.2.1.2.1.1

Factor $4$ out of $16$ .

$16 a^{2} + 4 (4) (\frac{a}{4}) + 16 (\frac{1}{16}) = 0$

Step 2.6.2.1.2.1.2

Cancel the common factor.

$16 a^{2} + 4 \cdot (4 (\frac{a}{4})) + 16 (\frac{1}{16}) = 0$

Step 2.6.2.1.2.1.3

Rewrite the expression.

$16 a^{2} + 4 a + 16 (\frac{1}{16}) = 0$

Step 2.6.2.1.2.2

Cancel the common factor of $16$ .

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Step 2.6.2.1.2.2.1

Cancel the common factor.

$16 a^{2} + 4 a + 16 (\frac{1}{16}) = 0$

Step 2.6.2.1.2.2.2

Rewrite the expression.

$16 a^{2} + 4 a + 1 = 0$

Step 2.6.2.2

Use the quadratic formula to find the solutions.

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

Step 2.6.2.3

Substitute the values $a = 16$ , $b = 4$ , and $c = 1$ into the quadratic formula and solve for $a$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (16 \cdot 1)}}{2 \cdot 16}$

Step 2.6.2.4

Simplify.

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Step 2.6.2.4.1

Simplify the numerator.

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Step 2.6.2.4.1.1

Raise $4$ to the power of $2$ .

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

Step 2.6.2.4.1.2

Multiply $- 4 \cdot 16 \cdot 1$ .

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Step 2.6.2.4.1.2.1

Multiply $- 4$ by $16$ .

$a = \frac{- 4 \pm \sqrt{16 - 64 \cdot 1}}{2 \cdot 16}$

Step 2.6.2.4.1.2.2

Multiply $- 64$ by $1$ .

$a = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 16}$

Step 2.6.2.4.1.3

Subtract $64$ from $16$ .

$a = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

Step 2.6.2.4.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$a = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

Step 2.6.2.4.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$a = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

Step 2.6.2.4.1.6

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

$a = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

Step 2.6.2.4.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.6.2.4.1.7.1

Factor $16$ out of $48$ .

$a = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 16}$

Step 2.6.2.4.1.7.2

Rewrite $16$ as $4^{2}$ .

$a = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 16}$

Step 2.6.2.4.1.8

Pull terms out from under the radical.

$a = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 16}$

Step 2.6.2.4.1.9

Move $4$ to the left of $i$ .

$a = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 16}$

Step 2.6.2.4.2

Multiply $2$ by $16$ .

$a = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Step 2.6.2.4.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

$a = \frac{- 1 \pm i \sqrt{3}}{8}$

Step 2.6.2.5

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

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Step 2.6.2.5.1

Simplify the numerator.

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Step 2.6.2.5.1.1

Raise $4$ to the power of $2$ .

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

Step 2.6.2.5.1.2

Multiply $- 4 \cdot 16 \cdot 1$ .

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Step 2.6.2.5.1.2.1

Multiply $- 4$ by $16$ .

$a = \frac{- 4 \pm \sqrt{16 - 64 \cdot 1}}{2 \cdot 16}$

Step 2.6.2.5.1.2.2

Multiply $- 64$ by $1$ .

$a = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 16}$

Step 2.6.2.5.1.3

Subtract $64$ from $16$ .

$a = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

Step 2.6.2.5.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$a = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

Step 2.6.2.5.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$a = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

Step 2.6.2.5.1.6

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

$a = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

Step 2.6.2.5.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.6.2.5.1.7.1

Factor $16$ out of $48$ .

$a = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 16}$

Step 2.6.2.5.1.7.2

Rewrite $16$ as $4^{2}$ .

$a = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 16}$

Step 2.6.2.5.1.8

Pull terms out from under the radical.

$a = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 16}$

Step 2.6.2.5.1.9

Move $4$ to the left of $i$ .

$a = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 16}$

Step 2.6.2.5.2

Multiply $2$ by $16$ .

$a = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Step 2.6.2.5.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

$a = \frac{- 1 \pm i \sqrt{3}}{8}$

Step 2.6.2.5.4

Change the $\pm$ to $+$ .

$a = \frac{- 1 + i \sqrt{3}}{8}$

Step 2.6.2.5.5

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

$a = \frac{- 1 \cdot 1 + i \sqrt{3}}{8}$

Step 2.6.2.5.6

Factor $- 1$ out of $i \sqrt{3}$ .

$a = \frac{- 1 \cdot 1 - (- i \sqrt{3})}{8}$

Step 2.6.2.5.7

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

$a = \frac{- 1 (1 - i \sqrt{3})}{8}$

Step 2.6.2.5.8

Move the negative in front of the fraction.

$a = - \frac{1 - i \sqrt{3}}{8}$

Step 2.6.2.6

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

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Step 2.6.2.6.1

Simplify the numerator.

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Step 2.6.2.6.1.1

Raise $4$ to the power of $2$ .

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

Step 2.6.2.6.1.2

Multiply $- 4 \cdot 16 \cdot 1$ .

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Step 2.6.2.6.1.2.1

Multiply $- 4$ by $16$ .

$a = \frac{- 4 \pm \sqrt{16 - 64 \cdot 1}}{2 \cdot 16}$

Step 2.6.2.6.1.2.2

Multiply $- 64$ by $1$ .

$a = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 16}$

Step 2.6.2.6.1.3

Subtract $64$ from $16$ .

$a = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

Step 2.6.2.6.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$a = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

Step 2.6.2.6.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$a = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

Step 2.6.2.6.1.6

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

$a = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

Step 2.6.2.6.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.6.2.6.1.7.1

Factor $16$ out of $48$ .

$a = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 16}$

Step 2.6.2.6.1.7.2

Rewrite $16$ as $4^{2}$ .

$a = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 16}$

Step 2.6.2.6.1.8

Pull terms out from under the radical.

$a = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 16}$

Step 2.6.2.6.1.9

Move $4$ to the left of $i$ .

$a = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 16}$

Step 2.6.2.6.2

Multiply $2$ by $16$ .

$a = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Step 2.6.2.6.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

$a = \frac{- 1 \pm i \sqrt{3}}{8}$

Step 2.6.2.6.4

Change the $\pm$ to $-$ .

$a = \frac{- 1 - i \sqrt{3}}{8}$

Step 2.6.2.6.5

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

$a = \frac{- 1 \cdot 1 - i \sqrt{3}}{8}$

Step 2.6.2.6.6

Factor $- 1$ out of $- i \sqrt{3}$ .

$a = \frac{- 1 \cdot 1 - (i \sqrt{3})}{8}$

Step 2.6.2.6.7

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

$a = \frac{- 1 (1 + i \sqrt{3})}{8}$

Step 2.6.2.6.8

Move the negative in front of the fraction.

$a = - \frac{1 + i \sqrt{3}}{8}$

Step 2.6.2.7

The final answer is the combination of both solutions.

$a = - \frac{1 - i \sqrt{3}}{8}, - \frac{1 + i \sqrt{3}}{8}$

Step 2.7

The final solution is all the values that make $(a - \frac{1}{4}) (a^{2} + \frac{a}{4} + \frac{1}{16}) = 0$ true.

$a = \frac{1}{4}, - \frac{1 - i \sqrt{3}}{8}, - \frac{1 + i \sqrt{3}}{8}$

Step 2.8

Remove all values that contain imaginary components.

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Step 2.8.1

There are no imaginary components. Add $\frac{1}{4}$ to the final answer.

$\frac{1}{4}$ is a real number

Step 2.8.2

The letter $i$ represents an imaginary component, and is not a real number. Do not add $- \frac{1 - i \sqrt{3}}{8}$ to the final answer.

$- \frac{1 - i \sqrt{3}}{8}$ is not a real number

Step 2.8.3

The letter $i$ represents an imaginary component, and is not a real number. Do not add $- \frac{1 + i \sqrt{3}}{8}$ to the final answer.

$- \frac{1 + i \sqrt{3}}{8}$ is not a real number

Step 2.8.4

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

$a = \frac{1}{4}$

Step 3

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

$f (x) = {(\frac{1}{4})}^{x}$