Algebra Examples

$(- 3, \frac{343}{64})$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $\frac{343}{64}$ of the point, and set $x$ to the $x$ value $- 3$ of the point.

$\frac{343}{64} = a^{- 3}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{- 3} = \frac{343}{64}$ .

$a^{- 3} = \frac{343}{64}$

Step 2.2

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

$\frac{1}{a^{3}} = \frac{343}{64}$

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.

$a^{3}, 64$

Step 2.3.2

Since $a^{3}, 64$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 64$ then find LCM for the variable part $a^{3}$ .

Step 2.3.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.3.5

The prime factors for $64$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 2.3.5.1

$64$ has factors of $2$ and $32$ .

$2 \cdot 32$

Step 2.3.5.2

$32$ has factors of $2$ and $16$ .

$2 \cdot 2 \cdot 16$

Step 2.3.5.3

$16$ has factors of $2$ and $8$ .

$2 \cdot 2 \cdot 2 \cdot 8$

Step 2.3.5.4

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 4$

Step 2.3.5.5

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

Step 2.3.6

Multiply $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 2.3.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

Step 2.3.6.2

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 2 \cdot 2$

Step 2.3.6.3

Multiply $8$ by $2$ .

$16 \cdot 2 \cdot 2$

Step 2.3.6.4

Multiply $16$ by $2$ .

$32 \cdot 2$

Step 2.3.6.5

Multiply $32$ by $2$ .

$64$

Step 2.3.7

The factors for $a^{3}$ are $a \cdot a \cdot a$ , which is $a$ multiplied by each other $3$ times.

$a^{3} = a \cdot a \cdot a$

$a$ occurs $3$ times.

Step 2.3.8

The LCM of $a^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$a \cdot a \cdot a$

Step 2.3.9

Simplify $a \cdot a \cdot a$ .

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Step 2.3.9.1

Multiply $a$ by $a$ .

$a^{2} \cdot a$

Step 2.3.9.2

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 2.3.9.2.1

Multiply $a^{2}$ by $a$ .

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Step 2.3.9.2.1.1

Raise $a$ to the power of $1$ .

$a^{2} \cdot a^{1}$

Step 2.3.9.2.1.2

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

$a^{2 + 1}$

Step 2.3.9.2.2

Add $2$ and $1$ .

$a^{3}$

Step 2.3.10

The LCM for $a^{3}, 64$ is the numeric part $64$ multiplied by the variable part.

$64 a^{3}$

Step 2.4

Multiply each term in $\frac{1}{a^{3}} = \frac{343}{64}$ by $64 a^{3}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{1}{a^{3}} = \frac{343}{64}$ by $64 a^{3}$ .

$\frac{1}{a^{3}} (64 a^{3}) = \frac{343}{64} (64 a^{3})$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Rewrite using the commutative property of multiplication.

$64 \frac{1}{a^{3}} a^{3} = \frac{343}{64} (64 a^{3})$

Step 2.4.2.2

Combine $64$ and $\frac{1}{a^{3}}$ .

$\frac{64}{a^{3}} a^{3} = \frac{343}{64} (64 a^{3})$

Step 2.4.2.3

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

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Step 2.4.2.3.1

Cancel the common factor.

$\frac{64}{a^{3}} a^{3} = \frac{343}{64} (64 a^{3})$

Step 2.4.2.3.2

Rewrite the expression.

$64 = \frac{343}{64} (64 a^{3})$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Cancel the common factor of $64$ .

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Step 2.4.3.1.1

Factor $64$ out of $64 a^{3}$ .

$64 = \frac{343}{64} (64 (a^{3}))$

Step 2.4.3.1.2

Cancel the common factor.

$64 = \frac{343}{64} (64 a^{3})$

Step 2.4.3.1.3

Rewrite the expression.

$64 = 343 a^{3}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $343 a^{3} = 64$ .

$343 a^{3} = 64$

Step 2.5.2

Subtract $64$ from both sides of the equation.

$343 a^{3} - 64 = 0$

Step 2.5.3

Factor the left side of the equation.

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Step 2.5.3.1

Rewrite $343 a^{3}$ as ${(7 a)}^{3}$ .

${(7 a)}^{3} - 64 = 0$

Step 2.5.3.2

Rewrite $64$ as $4^{3}$ .

${(7 a)}^{3} - 4^{3} = 0$

Step 2.5.3.3

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 $a = 7 a$ and $b = 4$ .

$(7 a - 4) ({(7 a)}^{2} + 7 a \cdot 4 + 4^{2}) = 0$

Step 2.5.3.4

Simplify.

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Step 2.5.3.4.1

Apply the product rule to $7 a$ .

$(7 a - 4) (7^{2} a^{2} + 7 a \cdot 4 + 4^{2}) = 0$

Step 2.5.3.4.2

Raise $7$ to the power of $2$ .

$(7 a - 4) (49 a^{2} + 7 a \cdot 4 + 4^{2}) = 0$

Step 2.5.3.4.3

Multiply $4$ by $7$ .

$(7 a - 4) (49 a^{2} + 28 a + 4^{2}) = 0$

Step 2.5.3.4.4

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

$(7 a - 4) (49 a^{2} + 28 a + 16) = 0$

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 $0$ .

$7 a - 4 = 0$

$49 a^{2} + 28 a + 16 = 0$

Step 2.5.5

Set $7 a - 4$ equal to $0$ and solve for $a$ .

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Step 2.5.5.1

Set $7 a - 4$ equal to $0$ .

$7 a - 4 = 0$

Step 2.5.5.2

Solve $7 a - 4 = 0$ for $a$ .

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Step 2.5.5.2.1

Add $4$ to both sides of the equation.

$7 a = 4$

Step 2.5.5.2.2

Divide each term in $7 a = 4$ by $7$ and simplify.

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Step 2.5.5.2.2.1

Divide each term in $7 a = 4$ by $7$ .

$\frac{7 a}{7} = \frac{4}{7}$

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 $7$ .

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Step 2.5.5.2.2.2.1.1

Cancel the common factor.

$\frac{7 a}{7} = \frac{4}{7}$

Step 2.5.5.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{4}{7}$

Step 2.5.6

Set $49 a^{2} + 28 a + 16$ equal to $0$ and solve for $a$ .

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Step 2.5.6.1

Set $49 a^{2} + 28 a + 16$ equal to $0$ .

$49 a^{2} + 28 a + 16 = 0$

Step 2.5.6.2

Solve $49 a^{2} + 28 a + 16 = 0$ 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 $a = 49$ , $b = 28$ , and $c = 16$ into the quadratic formula and solve for $a$ .

$\frac{- 28 \pm \sqrt{28^{2} - 4 \cdot (49 \cdot 16)}}{2 \cdot 49}$

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 $28$ to the power of $2$ .

$a = \frac{- 28 \pm \sqrt{784 - 4 \cdot 49 \cdot 16}}{2 \cdot 49}$

Step 2.5.6.2.3.1.2

Multiply $- 4 \cdot 49 \cdot 16$ .

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Step 2.5.6.2.3.1.2.1

Multiply $- 4$ by $49$ .

$a = \frac{- 28 \pm \sqrt{784 - 196 \cdot 16}}{2 \cdot 49}$

Step 2.5.6.2.3.1.2.2

Multiply $- 196$ by $16$ .

$a = \frac{- 28 \pm \sqrt{784 - 3136}}{2 \cdot 49}$

Step 2.5.6.2.3.1.3

Subtract $3136$ from $784$ .

$a = \frac{- 28 \pm \sqrt{- 2352}}{2 \cdot 49}$

Step 2.5.6.2.3.1.4

Rewrite $- 2352$ as $- 1 (2352)$ .

$a = \frac{- 28 \pm \sqrt{- 1 \cdot 2352}}{2 \cdot 49}$

Step 2.5.6.2.3.1.5

Rewrite $\sqrt{- 1 (2352)}$ as $\sqrt{- 1} \cdot \sqrt{2352}$ .

$a = \frac{- 28 \pm \sqrt{- 1} \cdot \sqrt{2352}}{2 \cdot 49}$

Step 2.5.6.2.3.1.6

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

$a = \frac{- 28 \pm i \cdot \sqrt{2352}}{2 \cdot 49}$

Step 2.5.6.2.3.1.7

Rewrite $2352$ as $28^{2} \cdot 3$ .

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Step 2.5.6.2.3.1.7.1

Factor $784$ out of $2352$ .

$a = \frac{- 28 \pm i \cdot \sqrt{784 (3)}}{2 \cdot 49}$

Step 2.5.6.2.3.1.7.2

Rewrite $784$ as $28^{2}$ .

$a = \frac{- 28 \pm i \cdot \sqrt{28^{2} \cdot 3}}{2 \cdot 49}$

Step 2.5.6.2.3.1.8

Pull terms out from under the radical.

$a = \frac{- 28 \pm i \cdot (28 \sqrt{3})}{2 \cdot 49}$

Step 2.5.6.2.3.1.9

Move $28$ to the left of $i$ .

$a = \frac{- 28 \pm 28 i \sqrt{3}}{2 \cdot 49}$

Step 2.5.6.2.3.2

Multiply $2$ by $49$ .

$a = \frac{- 28 \pm 28 i \sqrt{3}}{98}$

Step 2.5.6.2.3.3

Simplify $\frac{- 28 \pm 28 i \sqrt{3}}{98}$ .

$a = \frac{- 2 \pm 2 i \sqrt{3}}{7}$

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 $28$ to the power of $2$ .

$a = \frac{- 28 \pm \sqrt{784 - 4 \cdot 49 \cdot 16}}{2 \cdot 49}$

Step 2.5.6.2.4.1.2

Multiply $- 4 \cdot 49 \cdot 16$ .

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Step 2.5.6.2.4.1.2.1

Multiply $- 4$ by $49$ .

$a = \frac{- 28 \pm \sqrt{784 - 196 \cdot 16}}{2 \cdot 49}$

Step 2.5.6.2.4.1.2.2

Multiply $- 196$ by $16$ .

$a = \frac{- 28 \pm \sqrt{784 - 3136}}{2 \cdot 49}$

Step 2.5.6.2.4.1.3

Subtract $3136$ from $784$ .

$a = \frac{- 28 \pm \sqrt{- 2352}}{2 \cdot 49}$

Step 2.5.6.2.4.1.4

Rewrite $- 2352$ as $- 1 (2352)$ .

$a = \frac{- 28 \pm \sqrt{- 1 \cdot 2352}}{2 \cdot 49}$

Step 2.5.6.2.4.1.5

Rewrite $\sqrt{- 1 (2352)}$ as $\sqrt{- 1} \cdot \sqrt{2352}$ .

$a = \frac{- 28 \pm \sqrt{- 1} \cdot \sqrt{2352}}{2 \cdot 49}$

Step 2.5.6.2.4.1.6

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

$a = \frac{- 28 \pm i \cdot \sqrt{2352}}{2 \cdot 49}$

Step 2.5.6.2.4.1.7

Rewrite $2352$ as $28^{2} \cdot 3$ .

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Step 2.5.6.2.4.1.7.1

Factor $784$ out of $2352$ .

$a = \frac{- 28 \pm i \cdot \sqrt{784 (3)}}{2 \cdot 49}$

Step 2.5.6.2.4.1.7.2

Rewrite $784$ as $28^{2}$ .

$a = \frac{- 28 \pm i \cdot \sqrt{28^{2} \cdot 3}}{2 \cdot 49}$

Step 2.5.6.2.4.1.8

Pull terms out from under the radical.

$a = \frac{- 28 \pm i \cdot (28 \sqrt{3})}{2 \cdot 49}$

Step 2.5.6.2.4.1.9

Move $28$ to the left of $i$ .

$a = \frac{- 28 \pm 28 i \sqrt{3}}{2 \cdot 49}$

Step 2.5.6.2.4.2

Multiply $2$ by $49$ .

$a = \frac{- 28 \pm 28 i \sqrt{3}}{98}$

Step 2.5.6.2.4.3

Simplify $\frac{- 28 \pm 28 i \sqrt{3}}{98}$ .

$a = \frac{- 2 \pm 2 i \sqrt{3}}{7}$

Step 2.5.6.2.4.4

Change the $\pm$ to $+$ .

$a = \frac{- 2 + 2 i \sqrt{3}}{7}$

Step 2.5.6.2.4.5

Rewrite $- 2$ as $- 1 (2)$ .

$a = \frac{- 1 \cdot 2 + 2 i \sqrt{3}}{7}$

Step 2.5.6.2.4.6

Factor $- 1$ out of $2 i \sqrt{3}$ .

$a = \frac{- 1 \cdot 2 - (- 2 i \sqrt{3})}{7}$

Step 2.5.6.2.4.7

Factor $- 1$ out of $- 1 (2) - (- 2 i \sqrt{3})$ .

$a = \frac{- 1 (2 - 2 i \sqrt{3})}{7}$

Step 2.5.6.2.4.8

Move the negative in front of the fraction.

$a = - \frac{2 - 2 i \sqrt{3}}{7}$

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 $28$ to the power of $2$ .

$a = \frac{- 28 \pm \sqrt{784 - 4 \cdot 49 \cdot 16}}{2 \cdot 49}$

Step 2.5.6.2.5.1.2

Multiply $- 4 \cdot 49 \cdot 16$ .

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Step 2.5.6.2.5.1.2.1

Multiply $- 4$ by $49$ .

$a = \frac{- 28 \pm \sqrt{784 - 196 \cdot 16}}{2 \cdot 49}$

Step 2.5.6.2.5.1.2.2

Multiply $- 196$ by $16$ .

$a = \frac{- 28 \pm \sqrt{784 - 3136}}{2 \cdot 49}$

Step 2.5.6.2.5.1.3

Subtract $3136$ from $784$ .

$a = \frac{- 28 \pm \sqrt{- 2352}}{2 \cdot 49}$

Step 2.5.6.2.5.1.4

Rewrite $- 2352$ as $- 1 (2352)$ .

$a = \frac{- 28 \pm \sqrt{- 1 \cdot 2352}}{2 \cdot 49}$

Step 2.5.6.2.5.1.5

Rewrite $\sqrt{- 1 (2352)}$ as $\sqrt{- 1} \cdot \sqrt{2352}$ .

$a = \frac{- 28 \pm \sqrt{- 1} \cdot \sqrt{2352}}{2 \cdot 49}$

Step 2.5.6.2.5.1.6

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

$a = \frac{- 28 \pm i \cdot \sqrt{2352}}{2 \cdot 49}$

Step 2.5.6.2.5.1.7

Rewrite $2352$ as $28^{2} \cdot 3$ .

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Step 2.5.6.2.5.1.7.1

Factor $784$ out of $2352$ .

$a = \frac{- 28 \pm i \cdot \sqrt{784 (3)}}{2 \cdot 49}$

Step 2.5.6.2.5.1.7.2

Rewrite $784$ as $28^{2}$ .

$a = \frac{- 28 \pm i \cdot \sqrt{28^{2} \cdot 3}}{2 \cdot 49}$

Step 2.5.6.2.5.1.8

Pull terms out from under the radical.

$a = \frac{- 28 \pm i \cdot (28 \sqrt{3})}{2 \cdot 49}$

Step 2.5.6.2.5.1.9

Move $28$ to the left of $i$ .

$a = \frac{- 28 \pm 28 i \sqrt{3}}{2 \cdot 49}$

Step 2.5.6.2.5.2

Multiply $2$ by $49$ .

$a = \frac{- 28 \pm 28 i \sqrt{3}}{98}$

Step 2.5.6.2.5.3

Simplify $\frac{- 28 \pm 28 i \sqrt{3}}{98}$ .

$a = \frac{- 2 \pm 2 i \sqrt{3}}{7}$

Step 2.5.6.2.5.4

Change the $\pm$ to $-$ .

$a = \frac{- 2 - 2 i \sqrt{3}}{7}$

Step 2.5.6.2.5.5

Rewrite $- 2$ as $- 1 (2)$ .

$a = \frac{- 1 \cdot 2 - 2 i \sqrt{3}}{7}$

Step 2.5.6.2.5.6

Factor $- 1$ out of $- 2 i \sqrt{3}$ .

$a = \frac{- 1 \cdot 2 - (2 i \sqrt{3})}{7}$

Step 2.5.6.2.5.7

Factor $- 1$ out of $- 1 (2) - (2 i \sqrt{3})$ .

$a = \frac{- 1 (2 + 2 i \sqrt{3})}{7}$

Step 2.5.6.2.5.8

Move the negative in front of the fraction.

$a = - \frac{2 + 2 i \sqrt{3}}{7}$

Step 2.5.6.2.6

The final answer is the combination of both solutions.

$a = - \frac{2 - 2 i \sqrt{3}}{7}, - \frac{2 + 2 i \sqrt{3}}{7}$

Step 2.5.7

The final solution is all the values that make $(7 a - 4) (49 a^{2} + 28 a + 16) = 0$ true.

$a = \frac{4}{7}, - \frac{2 - 2 i \sqrt{3}}{7}, - \frac{2 + 2 i \sqrt{3}}{7}$

Step 2.6

Remove all values that contain imaginary components.

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Step 2.6.1

There are no imaginary components. Add $\frac{4}{7}$ to the final answer.

$\frac{4}{7}$ is a real number

Step 2.6.2

The letter $i$ represents an imaginary component, and is not a real number. Do not add $- \frac{2 - 2 i \sqrt{3}}{7}$ to the final answer.

$- \frac{2 - 2 i \sqrt{3}}{7}$ 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 $- \frac{2 + 2 i \sqrt{3}}{7}$ to the final answer.

$- \frac{2 + 2 i \sqrt{3}}{7}$ is not a real number

Step 2.6.4

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

$a = \frac{4}{7}$

Step 3

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

$f (x) = {(\frac{4}{7})}^{x}$