Basic Math Examples

$\frac{{(w + 5)}^{2}}{w - 5} \div w^{2} - 25 \div 5 w - 25$

Step 1

Simplify each term.

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Step 1.1

Rewrite the division as a fraction.

$\frac{\frac{{(w + 5)}^{2}}{w - 5}}{w^{2}} - 25 \div 5 w - 25$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{{(w + 5)}^{2}}{w - 5} \cdot \frac{1}{w^{2}} - 25 \div 5 w - 25$

Step 1.3

Multiply $\frac{{(w + 5)}^{2}}{w - 5}$ by $\frac{1}{w^{2}}$ .

$\frac{{(w + 5)}^{2}}{(w - 5) w^{2}} - 25 \div 5 w - 25$

Step 1.4

Rewrite the division as a fraction.

$\frac{{(w + 5)}^{2}}{(w - 5) w^{2}} - \frac{25}{5 w} - 25$

Step 1.5

Cancel the common factor of $25$ and $5$ .

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Step 1.5.1

Factor $5$ out of $25$ .

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $5$ out of $5 w$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

$\frac{{(w + 5)}^{2}}{(w - 5) w^{2}} - \frac{5}{w} - 25$

Step 2

To write $- \frac{5}{w}$ as a fraction with a common denominator, multiply by $\frac{(w - 5) w}{(w - 5) w}$ .

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

Step 3

Write each expression with a common denominator of $(w - 5) w^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{w}$ by $\frac{(w - 5) w}{(w - 5) w}$ .

$\frac{{(w + 5)}^{2}}{(w - 5) w^{2}} - \frac{5 ((w - 5) w)}{w ((w - 5) w)} - 25$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

$\frac{{(w + 5)}^{2}}{(w - 5) w^{2}} - \frac{5 ((w - 5) w)}{w^{2} (w - 5)} - 25$

Step 3.6

Reorder the factors of $(w - 5) w^{2}$ .

$\frac{{(w + 5)}^{2}}{w^{2} (w - 5)} - \frac{5 ((w - 5) w)}{w^{2} (w - 5)} - 25$

Step 4

Combine the numerators over the common denominator.

$\frac{{(w + 5)}^{2} - 5 ((w - 5) w)}{w^{2} (w - 5)} - 25$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${(w + 5)}^{2}$ as $(w + 5) (w + 5)$ .

$\frac{(w + 5) (w + 5) - 5 (w - 5) w}{w^{2} (w - 5)} - 25$

Step 5.2

Expand $(w + 5) (w + 5)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$\frac{w (w + 5) + 5 (w + 5) - 5 (w - 5) w}{w^{2} (w - 5)} - 25$

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $w$ by $w$ .

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

Step 5.3.1.2

Move $5$ to the left of $w$ .

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

Step 5.3.1.3

Multiply $5$ by $5$ .

$\frac{w^{2} + 5 w + 5 w + 25 - 5 (w - 5) w}{w^{2} (w - 5)} - 25$

Step 5.3.2

Add $5 w$ and $5 w$ .

$\frac{w^{2} + 10 w + 25 - 5 (w - 5) w}{w^{2} (w - 5)} - 25$

Step 5.4

Apply the distributive property.

$\frac{w^{2} + 10 w + 25 + (- 5 w - 5 \cdot - 5) w}{w^{2} (w - 5)} - 25$

Step 5.5

Multiply $- 5$ by $- 5$ .

$\frac{w^{2} + 10 w + 25 + (- 5 w + 25) w}{w^{2} (w - 5)} - 25$

Step 5.6

Apply the distributive property.

$\frac{w^{2} + 10 w + 25 - 5 w \cdot w + 25 w}{w^{2} (w - 5)} - 25$

Step 5.7

Multiply $w$ by $w$ by adding the exponents.

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Step 5.7.1

Move $w$ .

$\frac{w^{2} + 10 w + 25 - 5 (w \cdot w) + 25 w}{w^{2} (w - 5)} - 25$

Step 5.7.2

Multiply $w$ by $w$ .

$\frac{w^{2} + 10 w + 25 - 5 w^{2} + 25 w}{w^{2} (w - 5)} - 25$

Step 5.8

Subtract $5 w^{2}$ from $w^{2}$ .

$\frac{- 4 w^{2} + 10 w + 25 + 25 w}{w^{2} (w - 5)} - 25$

Step 5.9

Add $10 w$ and $25 w$ .

$\frac{- 4 w^{2} + 35 w + 25}{w^{2} (w - 5)} - 25$

Step 6

To write $- 25$ as a fraction with a common denominator, multiply by $\frac{(w - 5) w^{2}}{(w - 5) w^{2}}$ .

$\frac{- 4 w^{2} + 35 w + 25}{w^{2} (w - 5)} - 25 \cdot \frac{(w - 5) w^{2}}{(w - 5) w^{2}}$

Step 7

Write each expression with a common denominator of $(w - 5) w^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Combine $- 25$ and $\frac{(w - 5) w^{2}}{(w - 5) w^{2}}$ .

$\frac{- 4 w^{2} + 35 w + 25}{w^{2} (w - 5)} + \frac{- 25 ((w - 5) w^{2})}{(w - 5) w^{2}}$

Step 7.2

Reorder the factors of $(w - 5) w^{2}$ .

$\frac{- 4 w^{2} + 35 w + 25}{w^{2} (w - 5)} + \frac{- 25 ((w - 5) w^{2})}{w^{2} (w - 5)}$

Step 8

Combine the numerators over the common denominator.

$\frac{- 4 w^{2} + 35 w + 25 - 25 ((w - 5) w^{2})}{w^{2} (w - 5)}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{- 4 w^{2} + 35 w + 25 + (- 25 w - 25 \cdot - 5) w^{2}}{w^{2} (w - 5)}$

Step 9.2

Multiply $- 25$ by $- 5$ .

$\frac{- 4 w^{2} + 35 w + 25 + (- 25 w + 125) w^{2}}{w^{2} (w - 5)}$

Step 9.3

Apply the distributive property.

$\frac{- 4 w^{2} + 35 w + 25 - 25 w \cdot w^{2} + 125 w^{2}}{w^{2} (w - 5)}$

Step 9.4

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

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Step 9.4.1

Move $w^{2}$ .

$\frac{- 4 w^{2} + 35 w + 25 - 25 (w^{2} w) + 125 w^{2}}{w^{2} (w - 5)}$

Step 9.4.2

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

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Step 9.4.2.1

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

$\frac{- 4 w^{2} + 35 w + 25 - 25 (w^{2} w^{1}) + 125 w^{2}}{w^{2} (w - 5)}$

Step 9.4.2.2

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

$\frac{- 4 w^{2} + 35 w + 25 - 25 w^{2 + 1} + 125 w^{2}}{w^{2} (w - 5)}$

Step 9.4.3

Add $2$ and $1$ .

$\frac{- 4 w^{2} + 35 w + 25 - 25 w^{3} + 125 w^{2}}{w^{2} (w - 5)}$

Step 9.5

Add $- 4 w^{2}$ and $125 w^{2}$ .

$\frac{121 w^{2} + 35 w + 25 - 25 w^{3}}{w^{2} (w - 5)}$

Step 9.6

Reorder terms.

$\frac{- 25 w^{3} + 121 w^{2} + 35 w + 25}{w^{2} (w - 5)}$

Step 10

Simplify with factoring out.

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Step 10.1

Factor $- 1$ out of $- 25 w^{3}$ .

$\frac{- (25 w^{3}) + 121 w^{2} + 35 w + 25}{w^{2} (w - 5)}$

Step 10.2

Factor $- 1$ out of $121 w^{2}$ .

$\frac{- (25 w^{3}) - (- 121 w^{2}) + 35 w + 25}{w^{2} (w - 5)}$

Step 10.3

Factor $- 1$ out of $- (25 w^{3}) - (- 121 w^{2})$ .

$\frac{- (25 w^{3} - 121 w^{2}) + 35 w + 25}{w^{2} (w - 5)}$

Step 10.4

Factor $- 1$ out of $35 w$ .

$\frac{- (25 w^{3} - 121 w^{2}) - (- 35 w) + 25}{w^{2} (w - 5)}$

Step 10.5

Factor $- 1$ out of $- (25 w^{3} - 121 w^{2}) - (- 35 w)$ .

$\frac{- (25 w^{3} - 121 w^{2} - 35 w) + 25}{w^{2} (w - 5)}$

Step 10.6

Rewrite $25$ as $- 1 (- 25)$ .

$\frac{- (25 w^{3} - 121 w^{2} - 35 w) - 1 (- 25)}{w^{2} (w - 5)}$

Step 10.7

Factor $- 1$ out of $- (25 w^{3} - 121 w^{2} - 35 w) - 1 (- 25)$ .

$\frac{- (25 w^{3} - 121 w^{2} - 35 w - 25)}{w^{2} (w - 5)}$

Step 10.8

Simplify the expression.

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Step 10.8.1

Rewrite $- (25 w^{3} - 121 w^{2} - 35 w - 25)$ as $- 1 (25 w^{3} - 121 w^{2} - 35 w - 25)$ .

$\frac{- 1 (25 w^{3} - 121 w^{2} - 35 w - 25)}{w^{2} (w - 5)}$

Step 10.8.2

Move the negative in front of the fraction.

$- \frac{25 w^{3} - 121 w^{2} - 35 w - 25}{w^{2} (w - 5)}$