Basic Math Examples

$\frac{\sqrt{u} - 5}{5 + \sqrt{u}}$

Step 1

Multiply $\frac{\sqrt{u} - 5}{5 + \sqrt{u}}$ by $\frac{5 - \sqrt{u}}{5 - \sqrt{u}}$ .

$\frac{\sqrt{u} - 5}{5 + \sqrt{u}} \cdot \frac{5 - \sqrt{u}}{5 - \sqrt{u}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{u} - 5}{5 + \sqrt{u}}$ by $\frac{5 - \sqrt{u}}{5 - \sqrt{u}}$ .

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{(5 + \sqrt{u}) (5 - \sqrt{u})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{25 - 5 \sqrt{u} + \sqrt{u} \cdot 5 - {\sqrt{u}}^{2}}$

Step 2.3

Simplify.

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Step 2.3.1

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

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{25 - {(u^{\frac{1}{2}})}^{2}}$

Step 2.3.2

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

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{25 - u^{\frac{1}{2} \cdot 2}}$

Step 2.3.3

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

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{25 - u^{\frac{2}{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{25 - u^{\frac{2}{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{25 - u^{1}}$

Step 2.3.5

Simplify.

$\frac{(\sqrt{u} - 5) (5 - \sqrt{u})}{25 - u}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $- 1$ out of $\sqrt{u}$ .

$\frac{(- 1 (- \sqrt{u}) - 5) (5 - \sqrt{u})}{25 - u}$

Step 3.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{(- 1 (- \sqrt{u}) - 1 (5)) (5 - \sqrt{u})}{25 - u}$

Step 3.3

Factor $- 1$ out of $- 1 (- \sqrt{u}) - 1 (5)$ .

$\frac{- 1 (- \sqrt{u} + 5) (5 - \sqrt{u})}{25 - u}$

Step 3.4

Reorder terms.

$\frac{- 1 (5 - \sqrt{u}) (5 - \sqrt{u})}{25 - u}$

Step 3.5

Raise $5 - \sqrt{u}$ to the power of $1$ .

$\frac{- 1 ({(5 - \sqrt{u})}^{1} (5 - \sqrt{u}))}{25 - u}$

Step 3.6

Raise $5 - \sqrt{u}$ to the power of $1$ .

$\frac{- 1 ({(5 - \sqrt{u})}^{1} {(5 - \sqrt{u})}^{1})}{25 - u}$

Step 3.7

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

$\frac{- 1 {(5 - \sqrt{u})}^{1 + 1}}{25 - u}$

Step 3.8

Add $1$ and $1$ .

$\frac{- 1 {(5 - \sqrt{u})}^{2}}{25 - u}$

Step 4

Rewrite ${(5 - \sqrt{u})}^{2}$ as $(5 - \sqrt{u}) (5 - \sqrt{u})$ .

$\frac{- 1 ((5 - \sqrt{u}) (5 - \sqrt{u}))}{25 - u}$

Step 5

Expand $(5 - \sqrt{u}) (5 - \sqrt{u})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{- 1 (5 (5 - \sqrt{u}) - \sqrt{u} (5 - \sqrt{u}))}{25 - u}$

Step 5.2

Apply the distributive property.

$\frac{- 1 (5 \cdot 5 + 5 (- \sqrt{u}) - \sqrt{u} (5 - \sqrt{u}))}{25 - u}$

Step 5.3

Apply the distributive property.

$\frac{- 1 (5 \cdot 5 + 5 (- \sqrt{u}) - \sqrt{u} \cdot 5 - \sqrt{u} (- \sqrt{u}))}{25 - u}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $5$ by $5$ .

$\frac{- 1 (25 + 5 (- \sqrt{u}) - \sqrt{u} \cdot 5 - \sqrt{u} (- \sqrt{u}))}{25 - u}$

Step 6.1.2

Multiply $- 1$ by $5$ .

$\frac{- 1 (25 - 5 \sqrt{u} - \sqrt{u} \cdot 5 - \sqrt{u} (- \sqrt{u}))}{25 - u}$

Step 6.1.3

Multiply $5$ by $- 1$ .

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} - \sqrt{u} (- \sqrt{u}))}{25 - u}$

Step 6.1.4

Multiply $- \sqrt{u} (- \sqrt{u})$ .

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Step 6.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + 1 \sqrt{u} \sqrt{u})}{25 - u}$

Step 6.1.4.2

Multiply $\sqrt{u}$ by $1$ .

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + \sqrt{u} \sqrt{u})}{25 - u}$

Step 6.1.4.3

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

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + {\sqrt{u}}^{1} \sqrt{u})}{25 - u}$

Step 6.1.4.4

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

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + {\sqrt{u}}^{1} {\sqrt{u}}^{1})}{25 - u}$

Step 6.1.4.5

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

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + {\sqrt{u}}^{1 + 1})}{25 - u}$

Step 6.1.4.6

Add $1$ and $1$ .

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + {\sqrt{u}}^{2})}{25 - u}$

Step 6.1.5

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

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Step 6.1.5.1

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

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + {(u^{\frac{1}{2}})}^{2})}{25 - u}$

Step 6.1.5.2

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

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + u^{\frac{1}{2} \cdot 2})}{25 - u}$

Step 6.1.5.3

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

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + u^{\frac{2}{2}})}{25 - u}$

Step 6.1.5.4

Cancel the common factor of $2$ .

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Step 6.1.5.4.1

Cancel the common factor.

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + u^{\frac{2}{2}})}{25 - u}$

Step 6.1.5.4.2

Rewrite the expression.

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + u^{1})}{25 - u}$

Step 6.1.5.5

Simplify.

$\frac{- 1 (25 - 5 \sqrt{u} - 5 \sqrt{u} + u)}{25 - u}$

Step 6.2

Subtract $5 \sqrt{u}$ from $- 5 \sqrt{u}$ .

$\frac{- 1 (25 - 10 \sqrt{u} + u)}{25 - u}$

Step 7

Move the negative in front of the fraction.

$- \frac{25 - 10 \sqrt{u} + u}{25 - u}$