Precalculus Examples

$111100 = 110000 {(1 + \frac{r}{4})}^{4 (1.7)}$

Step 1

Rewrite the equation as $110000 {(1 + \frac{r}{4})}^{4 (1.7)} = 111100$ .

$110000 {(1 + \frac{r}{4})}^{4 (1.7)} = 111100$

Step 2

Multiply $4$ by $1.7$ .

$110000 {(1 + \frac{r}{4})}^{6.8} = 111100$

Step 3

Divide each term in $110000 {(1 + \frac{r}{4})}^{6.8} = 111100$ by $110000$ and simplify.

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Step 3.1

Divide each term in $110000 {(1 + \frac{r}{4})}^{6.8} = 111100$ by $110000$ .

$\frac{110000 {(1 + \frac{r}{4})}^{6.8}}{110000} = \frac{111100}{110000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $110000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{110000 {(1 + \frac{r}{4})}^{6.8}}{110000} = \frac{111100}{110000}$

Step 3.2.1.2

Divide ${(1 + \frac{r}{4})}^{6.8}$ by $1$ .

${(1 + \frac{r}{4})}^{6.8} = \frac{111100}{110000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $111100$ and $110000$ .

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Step 3.3.1.1

Factor $1100$ out of $111100$ .

${(1 + \frac{r}{4})}^{6.8} = \frac{1100 (101)}{110000}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $1100$ out of $110000$ .

${(1 + \frac{r}{4})}^{6.8} = \frac{1100 \cdot 101}{1100 \cdot 100}$

Step 3.3.1.2.2

Cancel the common factor.

${(1 + \frac{r}{4})}^{6.8} = \frac{1100 \cdot 101}{1100 \cdot 100}$

Step 3.3.1.2.3

Rewrite the expression.

${(1 + \frac{r}{4})}^{6.8} = \frac{101}{100}$

Step 4

Convert the decimal exponent to a fractional exponent.

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Step 4.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

${(1 + \frac{r}{4})}^{6 \frac{8}{10}} = \frac{101}{100}$

Step 4.2

Reduce the fractional part of the mixed number.

${(1 + \frac{r}{4})}^{6 \frac{4}{5}} = \frac{101}{100}$

Step 4.3

Convert $6 \frac{4}{5}$ to an improper fraction.

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Step 4.3.1

A mixed number is an addition of its whole and fractional parts.

${(1 + \frac{r}{4})}^{6 + \frac{4}{5}} = \frac{101}{100}$

Step 4.3.2

Add $6$ and $\frac{4}{5}$ .

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Step 4.3.2.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(1 + \frac{r}{4})}^{6 \cdot \frac{5}{5} + \frac{4}{5}} = \frac{101}{100}$

Step 4.3.2.2

Combine $6$ and $\frac{5}{5}$ .

${(1 + \frac{r}{4})}^{\frac{6 \cdot 5}{5} + \frac{4}{5}} = \frac{101}{100}$

Step 4.3.2.3

Combine the numerators over the common denominator.

${(1 + \frac{r}{4})}^{\frac{6 \cdot 5 + 4}{5}} = \frac{101}{100}$

Step 4.3.2.4

Simplify the numerator.

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Step 4.3.2.4.1

Multiply $6$ by $5$ .

${(1 + \frac{r}{4})}^{\frac{30 + 4}{5}} = \frac{101}{100}$

Step 4.3.2.4.2

Add $30$ and $4$ .

${(1 + \frac{r}{4})}^{\frac{34}{5}} = \frac{101}{100}$

Step 5

Raise each side of the equation to the power of $\frac{1}{6.8}$ to eliminate the fractional exponent on the left side.

${({(1 + \frac{r}{4})}^{\frac{34}{5}})}^{\frac{1}{6.8}} = {(\frac{101}{100})}^{\frac{1}{6.8}}$

Step 6

Simplify the exponent.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify ${({(1 + \frac{r}{4})}^{\frac{34}{5}})}^{\frac{1}{6.8}}$ .

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Step 6.1.1.1

Multiply the exponents in ${({(1 + \frac{r}{4})}^{\frac{34}{5}})}^{\frac{1}{6.8}}$ .

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Step 6.1.1.1.1

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

${(1 + \frac{r}{4})}^{\frac{34}{5} \cdot \frac{1}{6.8}} = {(\frac{101}{100})}^{\frac{1}{6.8}}$

Step 6.1.1.1.2

Cancel the common factor of $6.8$ .

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Step 6.1.1.1.2.1

Factor $6.8$ out of $34$ .

${(1 + \frac{r}{4})}^{\frac{6.8 (5)}{5} \cdot \frac{1}{6.8}} = {(\frac{101}{100})}^{\frac{1}{6.8}}$

Step 6.1.1.1.2.2

Cancel the common factor.

${(1 + \frac{r}{4})}^{\frac{6.8 \cdot 5}{5} \cdot \frac{1}{6.8}} = {(\frac{101}{100})}^{\frac{1}{6.8}}$

Step 6.1.1.1.2.3

Rewrite the expression.

${(1 + \frac{r}{4})}^{\frac{5}{5}} = {(\frac{101}{100})}^{\frac{1}{6.8}}$

Step 6.1.1.1.3

Divide $5$ by $5$ .

${(1 + \frac{r}{4})}^{1} = {(\frac{101}{100})}^{\frac{1}{6.8}}$

Step 6.1.1.2

Simplify.

$1 + \frac{r}{4} = {(\frac{101}{100})}^{\frac{1}{6.8}}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify ${(\frac{101}{100})}^{\frac{1}{6.8}}$ .

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Step 6.2.1.1

Divide $1$ by $6.8$ .

$1 + \frac{r}{4} = {(\frac{101}{100})}^{0.14705882}$

Step 6.2.1.2

Apply the product rule to $\frac{101}{100}$ .

$1 + \frac{r}{4} = \frac{101^{0.14705882}}{100^{0.14705882}}$

Step 6.2.1.3

Raise $101$ to the power of $0.14705882$ .

$1 + \frac{r}{4} = \frac{1.97130191}{100^{0.14705882}}$

Step 6.2.1.4

Raise $100$ to the power of $0.14705882$ .

$1 + \frac{r}{4} = \frac{1.97130191}{1.96841944}$

Step 6.2.1.5

Divide $1.97130191$ by $1.96841944$ .

$1 + \frac{r}{4} = 1.00146435$

Step 7

Solve for $r$ .

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Step 7.1

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

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Step 7.1.1

Subtract $1$ from both sides of the equation.

$\frac{r}{4} = 1.00146435 - 1$

Step 7.1.2

Subtract $1$ from $1.00146435$ .

$\frac{r}{4} = 0.00146435$

Step 7.2

Multiply both sides of the equation by $4$ .

$4 \frac{r}{4} = 4 \cdot 0.00146435$

Step 7.3

Simplify both sides of the equation.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Cancel the common factor of $4$ .

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Step 7.3.1.1.1

Cancel the common factor.

$4 \frac{r}{4} = 4 \cdot 0.00146435$

Step 7.3.1.1.2

Rewrite the expression.

$r = 4 \cdot 0.00146435$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Multiply $4$ by $0.00146435$ .

$r = 0.00585742$