Basic Math Examples

$y = 100 {(1 - r)}^{- 3}$

Step 1

Rewrite the equation as $100 {(1 - r)}^{- 3} = y$ .

$100 {(1 - r)}^{- 3} = y$

Step 2

Divide each term in $100 {(1 - r)}^{- 3} = y$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 {(1 - r)}^{- 3} = y$ by $100$ .

$\frac{100 {(1 - r)}^{- 3}}{100} = \frac{y}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Move ${(1 - r)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{100}{100 {(1 - r)}^{3}} = \frac{y}{100}$

Step 2.2.2

Cancel the common factor of $100$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{100}{100 {(1 - r)}^{3}} = \frac{y}{100}$

Step 2.2.2.2

Rewrite the expression.

$\frac{1}{{(1 - r)}^{3}} = \frac{y}{100}$

Step 3

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 100 = {(1 - r)}^{3} y$

Step 4

Solve the equation for $r$ .

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Step 4.1

Rewrite the equation as ${(1 - r)}^{3} y = 1 \cdot 100$ .

${(1 - r)}^{3} y = 1 \cdot 100$

Step 4.2

Multiply $100$ by $1$ .

${(1 - r)}^{3} y = 100$

Step 4.3

Divide each term in ${(1 - r)}^{3} y = 100$ by $y$ and simplify.

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Step 4.3.1

Divide each term in ${(1 - r)}^{3} y = 100$ by $y$ .

$\frac{{(1 - r)}^{3} y}{y} = \frac{100}{y}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $y$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{{(1 - r)}^{3} y}{y} = \frac{100}{y}$

Step 4.3.2.1.2

Divide ${(1 - r)}^{3}$ by $1$ .

${(1 - r)}^{3} = \frac{100}{y}$

Step 4.4

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

$1 - r = \sqrt[3]{\frac{100}{y}}$

Step 4.5

Simplify $\sqrt[3]{\frac{100}{y}}$ .

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Step 4.5.1

Rewrite $\sqrt[3]{\frac{100}{y}}$ as $\frac{\sqrt[3]{100}}{\sqrt[3]{y}}$ .

$1 - r = \frac{\sqrt[3]{100}}{\sqrt[3]{y}}$

Step 4.5.2

Multiply $\frac{\sqrt[3]{100}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$1 - r = \frac{\sqrt[3]{100}}{\sqrt[3]{y}} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$

Step 4.5.3

Combine and simplify the denominator.

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Step 4.5.3.1

Multiply $\frac{\sqrt[3]{100}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{\sqrt[3]{y} {\sqrt[3]{y}}^{2}}$

Step 4.5.3.2

Raise $\sqrt[3]{y}$ to the power of $1$ .

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2}}$

Step 4.5.3.3

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

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1 + 2}}$

Step 4.5.3.4

Add $1$ and $2$ .

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{3}}$

Step 4.5.3.5

Rewrite ${\sqrt[3]{y}}^{3}$ as $y$ .

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Step 4.5.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{{(y^{\frac{1}{3}})}^{3}}$

Step 4.5.3.5.2

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

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{y^{\frac{1}{3} \cdot 3}}$

Step 4.5.3.5.3

Combine $\frac{1}{3}$ and $3$ .

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Step 4.5.3.5.4

Cancel the common factor of $3$ .

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Step 4.5.3.5.4.1

Cancel the common factor.

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Step 4.5.3.5.4.2

Rewrite the expression.

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{y^{1}}$

Step 4.5.3.5.5

Simplify.

$1 - r = \frac{\sqrt[3]{100} {\sqrt[3]{y}}^{2}}{y}$

Step 4.5.4

Rewrite ${\sqrt[3]{y}}^{2}$ as $\sqrt[3]{y^{2}}$ .

$1 - r = \frac{\sqrt[3]{100} \sqrt[3]{y^{2}}}{y}$

Step 4.5.5

Combine using the product rule for radicals.

$1 - r = \frac{\sqrt[3]{100 y^{2}}}{y}$

Step 4.6

Subtract $1$ from both sides of the equation.

$- r = \frac{\sqrt[3]{100 y^{2}}}{y} - 1$

Step 4.7

Divide each term in $- r = \frac{\sqrt[3]{100 y^{2}}}{y} - 1$ by $- 1$ and simplify.

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Step 4.7.1

Divide each term in $- r = \frac{\sqrt[3]{100 y^{2}}}{y} - 1$ by $- 1$ .

$\frac{- r}{- 1} = \frac{\frac{\sqrt[3]{100 y^{2}}}{y}}{- 1} + \frac{- 1}{- 1}$

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Dividing two negative values results in a positive value.

$\frac{r}{1} = \frac{\frac{\sqrt[3]{100 y^{2}}}{y}}{- 1} + \frac{- 1}{- 1}$

Step 4.7.2.2

Divide $r$ by $1$ .

$r = \frac{\frac{\sqrt[3]{100 y^{2}}}{y}}{- 1} + \frac{- 1}{- 1}$

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Simplify each term.

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Step 4.7.3.1.1

Move the negative one from the denominator of $\frac{\frac{\sqrt[3]{100 y^{2}}}{y}}{- 1}$ .

$r = - 1 \cdot \frac{\sqrt[3]{100 y^{2}}}{y} + \frac{- 1}{- 1}$

Step 4.7.3.1.2

Rewrite $- 1 \cdot \frac{\sqrt[3]{100 y^{2}}}{y}$ as $- \frac{\sqrt[3]{100 y^{2}}}{y}$ .

$r = - \frac{\sqrt[3]{100 y^{2}}}{y} + \frac{- 1}{- 1}$

Step 4.7.3.1.3

Divide $- 1$ by $- 1$ .

$r = - \frac{\sqrt[3]{100 y^{2}}}{y} + 1$