Linear Algebra Examples

$5000 = 4000 {(1 + r (e f \cdot f))}^{4}$

Step 1

Since $r$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4000 {(1 + r e f \cdot f)}^{4} = 5000$

Step 2

Simplify.

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Step 2.1

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

$4000 {(1 + r e (f^{1} f))}^{4} = 5000$

Step 2.2

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

$4000 {(1 + r e (f^{1} f^{1}))}^{4} = 5000$

Step 2.3

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

$4000 {(1 + r e f^{1 + 1})}^{4} = 5000$

Step 2.4

Add $1$ and $1$ .

$4000 {(1 + r e f^{2})}^{4} = 5000$

Step 3

Divide each term in $4000 {(1 + r e f^{2})}^{4} = 5000$ by $4000$ and simplify.

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Step 3.1

Divide each term in $4000 {(1 + r e f^{2})}^{4} = 5000$ by $4000$ .

$\frac{4000 {(1 + r e f^{2})}^{4}}{4000} = \frac{5000}{4000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4000 {(1 + r e f^{2})}^{4}}{4000} = \frac{5000}{4000}$

Step 3.2.1.2

Divide ${(1 + r e f^{2})}^{4}$ by $1$ .

${(1 + r e f^{2})}^{4} = \frac{5000}{4000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $5000$ and $4000$ .

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Step 3.3.1.1

Factor $1000$ out of $5000$ .

${(1 + r e f^{2})}^{4} = \frac{1000 (5)}{4000}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $1000$ out of $4000$ .

${(1 + r e f^{2})}^{4} = \frac{1000 \cdot 5}{1000 \cdot 4}$

Step 3.3.1.2.2

Cancel the common factor.

${(1 + r e f^{2})}^{4} = \frac{1000 \cdot 5}{1000 \cdot 4}$

Step 3.3.1.2.3

Rewrite the expression.

${(1 + r e f^{2})}^{4} = \frac{5}{4}$

Step 4

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

$1 + r e f^{2} = \pm \sqrt[4]{\frac{5}{4}}$

Step 5

Simplify $\pm \sqrt[4]{\frac{5}{4}}$ .

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Step 5.1

Rewrite $\sqrt[4]{\frac{5}{4}}$ as $\frac{\sqrt[4]{5}}{\sqrt[4]{4}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5}}{\sqrt[4]{4}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

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

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5}}{\sqrt[4]{2^{2}}}$

Step 5.2.2

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5}}{\sqrt{\sqrt{2^{2}}}}$

Step 5.2.3

Pull terms out from under the radical, assuming positive real numbers.

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5}}{\sqrt{2}}$

Step 5.3

Multiply $\frac{\sqrt[4]{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

Multiply $\frac{\sqrt[4]{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.4.4

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

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.4.5

Add $1$ and $1$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.4.6.4

Cancel the common factor of $2$ .

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Step 5.4.6.4.1

Cancel the common factor.

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.4.6.4.2

Rewrite the expression.

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{2^{1}}$

Step 5.4.6.5

Evaluate the exponent.

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt{2}}{2}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Rewrite the expression using the least common index of $4$ .

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Step 5.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \cdot 2^{\frac{1}{2}}}{2}$

Step 5.5.1.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \cdot 2^{\frac{2}{4}}}{2}$

Step 5.5.1.3

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5} \sqrt[4]{2^{2}}}{2}$

Step 5.5.2

Combine using the product rule for radicals.

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5 \cdot 2^{2}}}{2}$

Step 5.5.3

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

$1 + r e f^{2} = \pm \frac{\sqrt[4]{5 \cdot 4}}{2}$

Step 5.6

Multiply $5$ by $4$ .

$1 + r e f^{2} = \pm \frac{\sqrt[4]{20}}{2}$

Step 6

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

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Step 6.1

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

$1 + r e f^{2} = \frac{\sqrt[4]{20}}{2}$

Step 6.2

Subtract $1$ from both sides of the equation.

$r e f^{2} = \frac{\sqrt[4]{20}}{2} - 1$

Step 6.3

Divide each term in $r e f^{2} = \frac{\sqrt[4]{20}}{2} - 1$ by $e f^{2}$ and simplify.

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Step 6.3.1

Divide each term in $r e f^{2} = \frac{\sqrt[4]{20}}{2} - 1$ by $e f^{2}$ .

$\frac{r e f^{2}}{e f^{2}} = \frac{\frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $e$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{r e f^{2}}{e f^{2}} = \frac{\frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.3.2.1.2

Rewrite the expression.

$\frac{r f^{2}}{f^{2}} = \frac{\frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.3.2.2

Cancel the common factor of $f^{2}$ .

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Step 6.3.2.2.1

Cancel the common factor.

$\frac{r f^{2}}{f^{2}} = \frac{\frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.3.2.2.2

Divide $r$ by $1$ .

$r = \frac{\frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify each term.

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Step 6.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$r = \frac{\sqrt[4]{20}}{2} \cdot \frac{1}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.3.3.1.2

Combine.

$r = \frac{\sqrt[4]{20} \cdot 1}{2 (e f^{2})} + \frac{- 1}{e f^{2}}$

Step 6.3.3.1.3

Multiply $\sqrt[4]{20}$ by $1$ .

$r = \frac{\sqrt[4]{20}}{2 e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.3.3.1.4

Move the negative in front of the fraction.

$r = \frac{\sqrt[4]{20}}{2 e f^{2}} - \frac{1}{e f^{2}}$

Step 6.4

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

$1 + r e f^{2} = - \frac{\sqrt[4]{20}}{2}$

Step 6.5

Subtract $1$ from both sides of the equation.

$r e f^{2} = - \frac{\sqrt[4]{20}}{2} - 1$

Step 6.6

Divide each term in $r e f^{2} = - \frac{\sqrt[4]{20}}{2} - 1$ by $e f^{2}$ and simplify.

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Step 6.6.1

Divide each term in $r e f^{2} = - \frac{\sqrt[4]{20}}{2} - 1$ by $e f^{2}$ .

$\frac{r e f^{2}}{e f^{2}} = \frac{- \frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.6.2

Simplify the left side.

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Step 6.6.2.1

Cancel the common factor of $e$ .

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Step 6.6.2.1.1

Cancel the common factor.

$\frac{r e f^{2}}{e f^{2}} = \frac{- \frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.6.2.1.2

Rewrite the expression.

$\frac{r f^{2}}{f^{2}} = \frac{- \frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.6.2.2

Cancel the common factor of $f^{2}$ .

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Step 6.6.2.2.1

Cancel the common factor.

$\frac{r f^{2}}{f^{2}} = \frac{- \frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.6.2.2.2

Divide $r$ by $1$ .

$r = \frac{- \frac{\sqrt[4]{20}}{2}}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.6.3

Simplify the right side.

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Step 6.6.3.1

Simplify each term.

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Step 6.6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$r = - \frac{\sqrt[4]{20}}{2} \cdot \frac{1}{e f^{2}} + \frac{- 1}{e f^{2}}$

Step 6.6.3.1.2

Multiply $\frac{1}{e f^{2}}$ by $\frac{\sqrt[4]{20}}{2}$ .

$r = - \frac{\sqrt[4]{20}}{e f^{2} \cdot 2} + \frac{- 1}{e f^{2}}$

Step 6.6.3.1.3

Move $2$ to the left of $e f^{2}$ .

$r = - \frac{\sqrt[4]{20}}{2 \cdot (e f^{2})} + \frac{- 1}{e f^{2}}$

Step 6.6.3.1.4

Move the negative in front of the fraction.

$r = - \frac{\sqrt[4]{20}}{2 e f^{2}} - \frac{1}{e f^{2}}$

Step 6.7

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

$r = \frac{\sqrt[4]{20}}{2 e f^{2}} - \frac{1}{e f^{2}}$

$r = - \frac{\sqrt[4]{20}}{2 e f^{2}} - \frac{1}{e f^{2}}$

$r = \frac{\sqrt[4]{20}}{2 e f^{2}} - \frac{1}{e f^{2}}$

$r = - \frac{\sqrt[4]{20}}{2 e f^{2}} - \frac{1}{e f^{2}}$

Step 7

Set the denominator in $\frac{\sqrt[4]{20}}{2 e f^{2}}$ equal to $0$ to find where the expression is undefined.

$2 e f^{2} = 0$

Step 8

Solve for $f$ .

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Step 8.1

Divide each term in $2 e f^{2} = 0$ by $2 e$ and simplify.

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Step 8.1.1

Divide each term in $2 e f^{2} = 0$ by $2 e$ .

$\frac{2 e f^{2}}{2 e} = \frac{0}{2 e}$

Step 8.1.2

Simplify the left side.

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Step 8.1.2.1

Cancel the common factor of $2$ .

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Step 8.1.2.1.1

Cancel the common factor.

$\frac{2 e f^{2}}{2 e} = \frac{0}{2 e}$

Step 8.1.2.1.2

Rewrite the expression.

$\frac{e f^{2}}{e} = \frac{0}{2 e}$

Step 8.1.2.2

Cancel the common factor of $e$ .

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Step 8.1.2.2.1

Cancel the common factor.

$\frac{e f^{2}}{e} = \frac{0}{2 e}$

Step 8.1.2.2.2

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

$f^{2} = \frac{0}{2 e}$

Step 8.1.3

Simplify the right side.

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Step 8.1.3.1

Cancel the common factor of $0$ and $2$ .

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Step 8.1.3.1.1

Factor $2$ out of $0$ .

$f^{2} = \frac{2 (0)}{2 e}$

Step 8.1.3.1.2

Cancel the common factors.

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Step 8.1.3.1.2.1

Factor $2$ out of $2 e$ .

$f^{2} = \frac{2 (0)}{2 (e)}$

Step 8.1.3.1.2.2

Cancel the common factor.

$f^{2} = \frac{2 \cdot 0}{2 e}$

Step 8.1.3.1.2.3

Rewrite the expression.

$f^{2} = \frac{0}{e}$

Step 8.1.3.2

Divide $0$ by $e$ .

$f^{2} = 0$

Step 8.2

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

$f = \pm \sqrt{0}$

Step 8.3

Simplify $\pm \sqrt{0}$ .

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Step 8.3.1

Rewrite $0$ as $0^{2}$ .

$f = \pm \sqrt{0^{2}}$

Step 8.3.2

Pull terms out from under the radical, assuming positive real numbers.

$f = \pm 0$

Step 8.3.3

Plus or minus $0$ is $0$ .

$f = 0$

Step 9

Set the denominator in $\frac{1}{e f^{2}}$ equal to $0$ to find where the expression is undefined.

$e f^{2} = 0$

Step 10

Solve for $f$ .

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Step 10.1

Divide each term in $e f^{2} = 0$ by $e$ and simplify.

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Step 10.1.1

Divide each term in $e f^{2} = 0$ by $e$ .

$\frac{e f^{2}}{e} = \frac{0}{e}$

Step 10.1.2

Simplify the left side.

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Step 10.1.2.1

Cancel the common factor of $e$ .

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Step 10.1.2.1.1

Cancel the common factor.

$\frac{e f^{2}}{e} = \frac{0}{e}$

Step 10.1.2.1.2

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

$f^{2} = \frac{0}{e}$

Step 10.1.3

Simplify the right side.

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Step 10.1.3.1

Divide $0$ by $e$ .

$f^{2} = 0$

Step 10.2

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

$f = \pm \sqrt{0}$

Step 10.3

Simplify $\pm \sqrt{0}$ .

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Step 10.3.1

Rewrite $0$ as $0^{2}$ .

$f = \pm \sqrt{0^{2}}$

Step 10.3.2

Pull terms out from under the radical, assuming positive real numbers.

$f = \pm 0$

Step 10.3.3

Plus or minus $0$ is $0$ .

$f = 0$

Step 11

The domain is all values of $f$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${f | f \neq 0}$

Step 12