Basic Math Examples

$(\sqrt{b} - \frac{b}{\sqrt{b} + 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{b}{\sqrt{b} + 1}$ by $\frac{\sqrt{b} - 1}{\sqrt{b} - 1}$ .

$(\sqrt{b} - (\frac{b}{\sqrt{b} + 1} \cdot \frac{\sqrt{b} - 1}{\sqrt{b} - 1})) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.2

Multiply $\frac{b}{\sqrt{b} + 1}$ by $\frac{\sqrt{b} - 1}{\sqrt{b} - 1}$ .

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{(\sqrt{b} + 1) (\sqrt{b} - 1)}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.3

Expand the denominator using the FOIL method.

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{{\sqrt{b}}^{2} + \sqrt{b} \cdot - 1 + \sqrt{b} - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.4

Simplify.

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Step 1.4.1

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

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{{(b^{\frac{1}{2}})}^{2} - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.4.2

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

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{b^{\frac{1}{2} \cdot 2} - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.4.3

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

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{b^{\frac{2}{2}} - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.4.4

Cancel the common factor of $2$ .

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Step 1.4.4.1

Cancel the common factor.

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{b^{\frac{2}{2}} - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.4.4.2

Rewrite the expression.

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{b^{1} - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 1.4.5

Simplify.

$(\sqrt{b} - \frac{b (\sqrt{b} - 1)}{b - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 2

To write $\sqrt{b}$ as a fraction with a common denominator, multiply by $\frac{b - 1}{b - 1}$ .

$(\frac{\sqrt{b} (b - 1)}{b - 1} - \frac{b (\sqrt{b} - 1)}{b - 1}) \cdot \frac{b - 1}{\sqrt{b}}$

Step 3

Combine the numerators over the common denominator.

$\frac{\sqrt{b} (b - 1) - b (\sqrt{b} - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{b^{\frac{1}{2}} (b - 1) - b (\sqrt{b} - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.2

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

$\frac{b^{\frac{1}{2}} (b - 1) - b (b^{\frac{1}{2}} - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.3

Factor $b^{\frac{1}{2}}$ out of $b^{\frac{1}{2}} (b - 1) - b (b^{\frac{1}{2}} - 1)$ .

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Step 4.3.1

Factor $b^{\frac{1}{2}}$ out of $- b (b^{\frac{1}{2}} - 1)$ .

$\frac{b^{\frac{1}{2}} (b - 1) + b^{\frac{1}{2}} (- b^{\frac{1}{2}} (b^{\frac{1}{2}} - 1))}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.3.2

Factor $b^{\frac{1}{2}}$ out of $b^{\frac{1}{2}} (b - 1) + b^{\frac{1}{2}} (- b^{\frac{1}{2}} (b^{\frac{1}{2}} - 1))$ .

$\frac{b^{\frac{1}{2}} (b - 1 - b^{\frac{1}{2}} (b^{\frac{1}{2}} - 1))}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.4

Apply the distributive property.

$\frac{b^{\frac{1}{2}} (b - 1 - b^{\frac{1}{2}} b^{\frac{1}{2}} - b^{\frac{1}{2}} \cdot - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.5

Multiply $b^{\frac{1}{2}}$ by $b^{\frac{1}{2}}$ by adding the exponents.

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Step 4.5.1

Move $b^{\frac{1}{2}}$ .

$\frac{b^{\frac{1}{2}} (b - 1 - (b^{\frac{1}{2}} b^{\frac{1}{2}}) - b^{\frac{1}{2}} \cdot - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.5.2

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

$\frac{b^{\frac{1}{2}} (b - 1 - b^{\frac{1}{2} + \frac{1}{2}} - b^{\frac{1}{2}} \cdot - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.5.3

Combine the numerators over the common denominator.

$\frac{b^{\frac{1}{2}} (b - 1 - b^{\frac{1 + 1}{2}} - b^{\frac{1}{2}} \cdot - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.5.4

Add $1$ and $1$ .

$\frac{b^{\frac{1}{2}} (b - 1 - b^{\frac{2}{2}} - b^{\frac{1}{2}} \cdot - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.5.5

Divide $2$ by $2$ .

$\frac{b^{\frac{1}{2}} (b - 1 - b^{1} - b^{\frac{1}{2}} \cdot - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.6

Simplify $- b^{1}$ .

$\frac{b^{\frac{1}{2}} (b - 1 - b - b^{\frac{1}{2}} \cdot - 1)}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.7

Multiply $- b^{\frac{1}{2}} \cdot - 1$ .

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Step 4.7.1

Multiply $- 1$ by $- 1$ .

$\frac{b^{\frac{1}{2}} (b - 1 - b + 1 b^{\frac{1}{2}})}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.7.2

Multiply $b^{\frac{1}{2}}$ by $1$ .

$\frac{b^{\frac{1}{2}} (b - 1 - b + b^{\frac{1}{2}})}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.8

Subtract $b$ from $b$ .

$\frac{b^{\frac{1}{2}} (0 - 1 + b^{\frac{1}{2}})}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 4.9

Subtract $1$ from $0$ .

$\frac{b^{\frac{1}{2}} (- 1 + b^{\frac{1}{2}})}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $b - 1$ .

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Step 5.1.1

Cancel the common factor.

$\frac{b^{\frac{1}{2}} (- 1 + b^{\frac{1}{2}})}{b - 1} \cdot \frac{b - 1}{\sqrt{b}}$

Step 5.1.2

Rewrite the expression.

$b^{\frac{1}{2}} (- 1 + b^{\frac{1}{2}}) \cdot \frac{1}{\sqrt{b}}$

Step 5.2

Combine $\frac{1}{\sqrt{b}}$ and $b^{\frac{1}{2}}$ .

$\frac{b^{\frac{1}{2}}}{\sqrt{b}} (- 1 + b^{\frac{1}{2}})$

Step 6

Multiply $\frac{b^{\frac{1}{2}}}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{b^{\frac{1}{2}}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} (- 1 + b^{\frac{1}{2}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{b^{\frac{1}{2}}}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{b^{\frac{1}{2}} \sqrt{b}}{\sqrt{b} \sqrt{b}} (- 1 + b^{\frac{1}{2}})$

Step 7.2

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

$\frac{b^{\frac{1}{2}} \sqrt{b}}{{\sqrt{b}}^{1} \sqrt{b}} (- 1 + b^{\frac{1}{2}})$

Step 7.3

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

$\frac{b^{\frac{1}{2}} \sqrt{b}}{{\sqrt{b}}^{1} {\sqrt{b}}^{1}} (- 1 + b^{\frac{1}{2}})$

Step 7.4

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

$\frac{b^{\frac{1}{2}} \sqrt{b}}{{\sqrt{b}}^{1 + 1}} (- 1 + b^{\frac{1}{2}})$

Step 7.5

Add $1$ and $1$ .

$\frac{b^{\frac{1}{2}} \sqrt{b}}{{\sqrt{b}}^{2}} (- 1 + b^{\frac{1}{2}})$

Step 7.6

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

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Step 7.6.1

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

$\frac{b^{\frac{1}{2}} \sqrt{b}}{{(b^{\frac{1}{2}})}^{2}} (- 1 + b^{\frac{1}{2}})$

Step 7.6.2

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

$\frac{b^{\frac{1}{2}} \sqrt{b}}{b^{\frac{1}{2} \cdot 2}} (- 1 + b^{\frac{1}{2}})$

Step 7.6.3

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

$\frac{b^{\frac{1}{2}} \sqrt{b}}{b^{\frac{2}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{b^{\frac{1}{2}} \sqrt{b}}{b^{\frac{2}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 7.6.4.2

Rewrite the expression.

$\frac{b^{\frac{1}{2}} \sqrt{b}}{b^{1}} (- 1 + b^{\frac{1}{2}})$

Step 7.6.5

Simplify.

$\frac{b^{\frac{1}{2}} \sqrt{b}}{b} (- 1 + b^{\frac{1}{2}})$

Step 8

Move $b^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{\sqrt{b}}{b \cdot b^{- \frac{1}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 9

Multiply $b$ by $b^{- \frac{1}{2}}$ by adding the exponents.

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Step 9.1

Multiply $b$ by $b^{- \frac{1}{2}}$ .

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Step 9.1.1

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

$\frac{\sqrt{b}}{b^{1} b^{- \frac{1}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 9.1.2

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

$\frac{\sqrt{b}}{b^{1 - \frac{1}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 9.2

Write $1$ as a fraction with a common denominator.

$\frac{\sqrt{b}}{b^{\frac{2}{2} - \frac{1}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 9.3

Combine the numerators over the common denominator.

$\frac{\sqrt{b}}{b^{\frac{2 - 1}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 9.4

Subtract $1$ from $2$ .

$\frac{\sqrt{b}}{b^{\frac{1}{2}}} (- 1 + b^{\frac{1}{2}})$

Step 10

Multiply $\frac{\sqrt{b}}{b^{\frac{1}{2}}}$ by $- 1 + b^{\frac{1}{2}}$ .

$\frac{\sqrt{b} (- 1 + b^{\frac{1}{2}})}{b^{\frac{1}{2}}}$

Step 11

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{\sqrt{b} (- 1 (1) + b^{\frac{1}{2}})}{b^{\frac{1}{2}}}$

Step 12

Factor $- 1$ out of $b^{\frac{1}{2}}$ .

$\frac{\sqrt{b} (- 1 (1) - 1 (- b^{\frac{1}{2}}))}{b^{\frac{1}{2}}}$

Step 13

Factor $- 1$ out of $- 1 (1) - 1 (- b^{\frac{1}{2}})$ .

$\frac{\sqrt{b} (- 1 (1 - b^{\frac{1}{2}}))}{b^{\frac{1}{2}}}$

Step 14

Move the negative in front of the fraction.

$- \frac{\sqrt{b} (1 - b^{\frac{1}{2}})}{b^{\frac{1}{2}}}$