Basic Math Examples

$4270 = 4000 {(1 + y)}^{10}$

Step 1

Rewrite the equation as $4000 {(1 + y)}^{10} = 4270$ .

$4000 {(1 + y)}^{10} = 4270$

Step 2

Divide each term in $4000 {(1 + y)}^{10} = 4270$ by $4000$ and simplify.

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Step 2.1

Divide each term in $4000 {(1 + y)}^{10} = 4270$ by $4000$ .

$\frac{4000 {(1 + y)}^{10}}{4000} = \frac{4270}{4000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4000 {(1 + y)}^{10}}{4000} = \frac{4270}{4000}$

Step 2.2.1.2

Divide ${(1 + y)}^{10}$ by $1$ .

${(1 + y)}^{10} = \frac{4270}{4000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $4270$ and $4000$ .

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Step 2.3.1.1

Factor $10$ out of $4270$ .

${(1 + y)}^{10} = \frac{10 (427)}{4000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $10$ out of $4000$ .

${(1 + y)}^{10} = \frac{10 \cdot 427}{10 \cdot 400}$

Step 2.3.1.2.2

Cancel the common factor.

${(1 + y)}^{10} = \frac{10 \cdot 427}{10 \cdot 400}$

Step 2.3.1.2.3

Rewrite the expression.

${(1 + y)}^{10} = \frac{427}{400}$

Step 3

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

$1 + y = \pm \sqrt[10]{\frac{427}{400}}$

Step 4

Simplify $\pm \sqrt[10]{\frac{427}{400}}$ .

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Step 4.1

Rewrite $\sqrt[10]{\frac{427}{400}}$ as $\frac{\sqrt[10]{427}}{\sqrt[10]{400}}$ .

$1 + y = \pm \frac{\sqrt[10]{427}}{\sqrt[10]{400}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Rewrite $400$ as $20^{2}$ .

$1 + y = \pm \frac{\sqrt[10]{427}}{\sqrt[10]{20^{2}}}$

Step 4.2.2

Rewrite $\sqrt[10]{20^{2}}$ as $\sqrt[5]{\sqrt{20^{2}}}$ .

$1 + y = \pm \frac{\sqrt[10]{427}}{\sqrt[5]{\sqrt{20^{2}}}}$

Step 4.2.3

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

$1 + y = \pm \frac{\sqrt[10]{427}}{\sqrt[5]{20}}$

Step 4.3

Multiply $\frac{\sqrt[10]{427}}{\sqrt[5]{20}}$ by $\frac{{\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{4}}$ .

$1 + y = \pm \frac{\sqrt[10]{427}}{\sqrt[5]{20}} \cdot \frac{{\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{4}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{\sqrt[10]{427}}{\sqrt[5]{20}}$ by $\frac{{\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{4}}$ .

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{\sqrt[5]{20} {\sqrt[5]{20}}^{4}}$

Step 4.4.2

Raise $\sqrt[5]{20}$ to the power of $1$ .

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{1} {\sqrt[5]{20}}^{4}}$

Step 4.4.3

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

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{1 + 4}}$

Step 4.4.4

Add $1$ and $4$ .

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{5}}$

Step 4.4.5

Rewrite ${\sqrt[5]{20}}^{5}$ as $20$ .

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Step 4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{20}$ as $20^{\frac{1}{5}}$ .

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{{(20^{\frac{1}{5}})}^{5}}$

Step 4.4.5.2

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

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{20^{\frac{1}{5} \cdot 5}}$

Step 4.4.5.3

Combine $\frac{1}{5}$ and $5$ .

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{20^{\frac{5}{5}}}$

Step 4.4.5.4

Cancel the common factor of $5$ .

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Step 4.4.5.4.1

Cancel the common factor.

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{20^{\frac{5}{5}}}$

Step 4.4.5.4.2

Rewrite the expression.

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{20^{1}}$

Step 4.4.5.5

Evaluate the exponent.

$1 + y = \pm \frac{\sqrt[10]{427} {\sqrt[5]{20}}^{4}}{20}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Rewrite ${\sqrt[5]{20}}^{4}$ as $\sqrt[5]{20^{4}}$ .

$1 + y = \pm \frac{\sqrt[10]{427} \sqrt[5]{20^{4}}}{20}$

Step 4.5.2

Raise $20$ to the power of $4$ .

$1 + y = \pm \frac{\sqrt[10]{427} \sqrt[5]{160000}}{20}$

Step 4.5.3

Rewrite $160000$ as $2^{5} \cdot 5000$ .

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Step 4.5.3.1

Factor $32$ out of $160000$ .

$1 + y = \pm \frac{\sqrt[10]{427} \sqrt[5]{32 (5000)}}{20}$

Step 4.5.3.2

Rewrite $32$ as $2^{5}$ .

$1 + y = \pm \frac{\sqrt[10]{427} \sqrt[5]{2^{5} \cdot 5000}}{20}$

Step 4.5.4

Pull terms out from under the radical.

$1 + y = \pm \frac{\sqrt[10]{427} \cdot 2 \sqrt[5]{5000}}{20}$

Step 4.5.5

Combine exponents.

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Step 4.5.5.1

Rewrite the expression using the least common index of $10$ .

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Step 4.5.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{5000}$ as $5000^{\frac{1}{5}}$ .

$1 + y = \pm \frac{2 (\sqrt[10]{427} \cdot 5000^{\frac{1}{5}})}{20}$

Step 4.5.5.1.2

Rewrite $5000^{\frac{1}{5}}$ as $5000^{\frac{2}{10}}$ .

$1 + y = \pm \frac{2 (\sqrt[10]{427} \cdot 5000^{\frac{2}{10}})}{20}$

Step 4.5.5.1.3

Rewrite $5000^{\frac{2}{10}}$ as $\sqrt[10]{5000^{2}}$ .

$1 + y = \pm \frac{2 (\sqrt[10]{427} \sqrt[10]{5000^{2}})}{20}$

Step 4.5.5.2

Combine using the product rule for radicals.

$1 + y = \pm \frac{2 \sqrt[10]{427 \cdot 5000^{2}}}{20}$

Step 4.5.5.3

Raise $5000$ to the power of $2$ .

$1 + y = \pm \frac{2 \sqrt[10]{427 \cdot 25000000}}{20}$

Step 4.5.5.4

Multiply $427$ by $25000000$ .

$1 + y = \pm \frac{2 \sqrt[10]{10675000000}}{20}$

Step 4.6

Cancel the common factor of $2$ and $20$ .

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Step 4.6.1

Factor $2$ out of $2 \sqrt[10]{10675000000}$ .

$1 + y = \pm \frac{2 (\sqrt[10]{10675000000})}{20}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $2$ out of $20$ .

$1 + y = \pm \frac{2 \sqrt[10]{10675000000}}{2 \cdot 10}$

Step 4.6.2.2

Cancel the common factor.

$1 + y = \pm \frac{2 \sqrt[10]{10675000000}}{2 \cdot 10}$

Step 4.6.2.3

Rewrite the expression.

$1 + y = \pm \frac{\sqrt[10]{10675000000}}{10}$

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.

$1 + y = \frac{\sqrt[10]{10675000000}}{10}$

Step 5.2

Subtract $1$ from both sides of the equation.

$y = \frac{\sqrt[10]{10675000000}}{10} - 1$

Step 5.3

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

$1 + y = - \frac{\sqrt[10]{10675000000}}{10}$

Step 5.4

Subtract $1$ from both sides of the equation.

$y = - \frac{\sqrt[10]{10675000000}}{10} - 1$

Step 5.5

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

$y = \frac{\sqrt[10]{10675000000}}{10} - 1, - \frac{\sqrt[10]{10675000000}}{10} - 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\sqrt[10]{10675000000}}{10} - 1, - \frac{\sqrt[10]{10675000000}}{10} - 1$

Decimal Form:

$y = 0.00655332 \dots, - 2.00655332 \dots$