Precalculus Examples

$4899.78 {(1 + x)}^{\frac{32}{360}} = 5000$

Step 1

Reduce the expression $\frac{32}{360}$ by cancelling the common factors.

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Step 1.1

Factor $8$ out of $32$ .

$4899.78 {(1 + x)}^{\frac{8 (4)}{360}} = 5000$

Step 1.2

Factor $8$ out of $360$ .

$4899.78 {(1 + x)}^{\frac{8 \cdot 4}{8 \cdot 45}} = 5000$

Step 1.3

Cancel the common factor.

$4899.78 {(1 + x)}^{\frac{8 \cdot 4}{8 \cdot 45}} = 5000$

Step 1.4

Rewrite the expression.

$4899.78 {(1 + x)}^{\frac{4}{45}} = 5000$

Step 2

Divide each term in $4899.78 {(1 + x)}^{\frac{4}{45}} = 5000$ by $4899.78$ and simplify.

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Step 2.1

Divide each term in $4899.78 {(1 + x)}^{\frac{4}{45}} = 5000$ by $4899.78$ .

$\frac{4899.78 {(1 + x)}^{\frac{4}{45}}}{4899.78} = \frac{5000}{4899.78}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor.

$\frac{4899.78 {(1 + x)}^{\frac{4}{45}}}{4899.78} = \frac{5000}{4899.78}$

Step 2.2.2

Divide ${(1 + x)}^{\frac{4}{45}}$ by $1$ .

${(1 + x)}^{\frac{4}{45}} = \frac{5000}{4899.78}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $5000$ by $4899.78$ .

${(1 + x)}^{\frac{4}{45}} = 1.02045397$

Step 3

Raise each side of the equation to the power of $\frac{45}{4}$ to eliminate the fractional exponent on the left side.

${({(1 + x)}^{\frac{4}{45}})}^{\frac{45}{4}} = \pm {1.02045397}^{\frac{45}{4}}$

Step 4

Simplify the exponent.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify ${({(1 + x)}^{\frac{4}{45}})}^{\frac{45}{4}}$ .

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Step 4.1.1.1

Multiply the exponents in ${({(1 + x)}^{\frac{4}{45}})}^{\frac{45}{4}}$ .

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Step 4.1.1.1.1

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

${(1 + x)}^{\frac{4}{45} \cdot \frac{45}{4}} = \pm {1.02045397}^{\frac{45}{4}}$

Step 4.1.1.1.2

Cancel the common factor of $4$ .

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Step 4.1.1.1.2.1

Cancel the common factor.

${(1 + x)}^{\frac{4}{45} \cdot \frac{45}{4}} = \pm {1.02045397}^{\frac{45}{4}}$

Step 4.1.1.1.2.2

Rewrite the expression.

${(1 + x)}^{\frac{1}{45} \cdot 45} = \pm {1.02045397}^{\frac{45}{4}}$

Step 4.1.1.1.3

Cancel the common factor of $45$ .

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Step 4.1.1.1.3.1

Cancel the common factor.

${(1 + x)}^{\frac{1}{45} \cdot 45} = \pm {1.02045397}^{\frac{45}{4}}$

Step 4.1.1.1.3.2

Rewrite the expression.

${(1 + x)}^{1} = \pm {1.02045397}^{\frac{45}{4}}$

Step 4.1.1.2

Simplify.

$1 + x = \pm {1.02045397}^{\frac{45}{4}}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Raise $1.02045397$ to the power of $\frac{45}{4}$ .

$1 + x = \pm 1.25581601$

Step 5

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

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Step 5.1

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

$1 + x = 1.25581601$

Step 5.2

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

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$x = 1.25581601 - 1$

Step 5.2.2

Subtract $1$ from $1.25581601$ .

$x = 0.25581601$

Step 5.3

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

$1 + x = - 1.25581601$

Step 5.4

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

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Step 5.4.1

Subtract $1$ from both sides of the equation.

$x = - 1.25581601 - 1$

Step 5.4.2

Subtract $1$ from $- 1.25581601$ .

$x = - 2.25581601$

Step 5.5

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

$x = 0.25581601, - 2.25581601$

Step 6

Exclude the solutions that do not make $4899.78 {(1 + x)}^{\frac{32}{360}} = 5000$ true.

$x = 0.25581601$