Basic Math Examples

$u - \frac{1}{u} \cdot 1 - \frac{1}{u^{2}}$

Step 1

Multiply $- 1$ by $1$ .

$u - \frac{1}{u} - \frac{1}{u^{2}}$

Step 2

To write $u$ as a fraction with a common denominator, multiply by $\frac{u}{u}$ .

$\frac{u \cdot u}{u} - \frac{1}{u} - \frac{1}{u^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{u \cdot u - 1}{u} - \frac{1}{u^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{u^{1} u - 1}{u} - \frac{1}{u^{2}}$

Step 4.2

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

$\frac{u^{1} u^{1} - 1}{u} - \frac{1}{u^{2}}$

Step 4.3

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

$\frac{u^{1 + 1} - 1}{u} - \frac{1}{u^{2}}$

Step 4.4

Add $1$ and $1$ .

$\frac{u^{2} - 1}{u} - \frac{1}{u^{2}}$

Step 4.5

Rewrite $1$ as $1^{2}$ .

$\frac{u^{2} - 1^{2}}{u} - \frac{1}{u^{2}}$

Step 4.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u$ and $b = 1$ .

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

Step 5

To write $\frac{(u + 1) (u - 1)}{u}$ as a fraction with a common denominator, multiply by $\frac{u}{u}$ .

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

Step 6

Write each expression with a common denominator of $u^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{(u + 1) (u - 1)}{u}$ by $\frac{u}{u}$ .

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

Step 6.2

Multiply $u$ by $u$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Expand $(u + 1) (u - 1)$ using the FOIL Method.

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Step 8.1.1

Apply the distributive property.

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

Step 8.1.2

Apply the distributive property.

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

Step 8.1.3

Apply the distributive property.

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

Step 8.2

Simplify and combine like terms.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $u$ by $u$ .

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

Step 8.2.1.2

Move $- 1$ to the left of $u$ .

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

Step 8.2.1.3

Rewrite $- 1 u$ as $- u$ .

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

Step 8.2.1.4

Multiply $u$ by $1$ .

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

Step 8.2.1.5

Multiply $- 1$ by $1$ .

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

Step 8.2.2

Add $- u$ and $u$ .

$\frac{(u^{2} + 0 - 1) u - 1}{u^{2}}$

Step 8.2.3

Add $u^{2}$ and $0$ .

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

Step 8.3

Apply the distributive property.

$\frac{u^{2} u - 1 u - 1}{u^{2}}$

Step 8.4

Multiply $u^{2}$ by $u$ by adding the exponents.

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Step 8.4.1

Multiply $u^{2}$ by $u$ .

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Step 8.4.1.1

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

$\frac{u^{2} u^{1} - 1 u - 1}{u^{2}}$

Step 8.4.1.2

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

$\frac{u^{2 + 1} - 1 u - 1}{u^{2}}$

Step 8.4.2

Add $2$ and $1$ .

$\frac{u^{3} - 1 u - 1}{u^{2}}$

Step 8.5

Rewrite $- 1 u$ as $- u$ .

$\frac{u^{3} - u - 1}{u^{2}}$