Pre-Algebra Examples

$\frac{5 n + 5}{5 n^{2} + 35 n - 40} + \frac{7 n}{3 n}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $5 n + 5$ and $5 n^{2} + 35 n - 40$ .

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Step 1.1.1

Factor $5$ out of $5 n$ .

$\frac{5 (n) + 5}{5 n^{2} + 35 n - 40} + \frac{7 n}{3 n}$

Step 1.1.2

Factor $5$ out of $5$ .

$\frac{5 (n) + 5 \cdot 1}{5 n^{2} + 35 n - 40} + \frac{7 n}{3 n}$

Step 1.1.3

Factor $5$ out of $5 (n) + 5 (1)$ .

$\frac{5 (n + 1)}{5 n^{2} + 35 n - 40} + \frac{7 n}{3 n}$

Step 1.1.4

Cancel the common factors.

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Step 1.1.4.1

Factor $5$ out of $5 n^{2}$ .

$\frac{5 (n + 1)}{5 (n^{2}) + 35 n - 40} + \frac{7 n}{3 n}$

Step 1.1.4.2

Factor $5$ out of $35 n$ .

$\frac{5 (n + 1)}{5 (n^{2}) + 5 (7 n) - 40} + \frac{7 n}{3 n}$

Step 1.1.4.3

Factor $5$ out of $5 (n^{2}) + 5 (7 n)$ .

$\frac{5 (n + 1)}{5 (n^{2} + 7 n) - 40} + \frac{7 n}{3 n}$

Step 1.1.4.4

Factor $5$ out of $- 40$ .

$\frac{5 (n + 1)}{5 (n^{2} + 7 n) + 5 \cdot - 8} + \frac{7 n}{3 n}$

Step 1.1.4.5

Factor $5$ out of $5 (n^{2} + 7 n) + 5 \cdot - 8$ .

$\frac{5 (n + 1)}{5 (n^{2} + 7 n - 8)} + \frac{7 n}{3 n}$

Step 1.1.4.6

Cancel the common factor.

$\frac{5 (n + 1)}{5 (n^{2} + 7 n - 8)} + \frac{7 n}{3 n}$

Step 1.1.4.7

Rewrite the expression.

$\frac{n + 1}{n^{2} + 7 n - 8} + \frac{7 n}{3 n}$

Step 1.2

Factor $n^{2} + 7 n - 8$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $7$ .

$- 1, 8$

Step 1.2.2

Write the factored form using these integers.

$\frac{n + 1}{(n - 1) (n + 8)} + \frac{7 n}{3 n}$

Step 1.3

Cancel the common factor of $n$ .

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Step 1.3.1

Cancel the common factor.

$\frac{n + 1}{(n - 1) (n + 8)} + \frac{7 n}{3 n}$

Step 1.3.2

Rewrite the expression.

$\frac{n + 1}{(n - 1) (n + 8)} + \frac{7}{3}$

Step 2

To write $\frac{n + 1}{(n - 1) (n + 8)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{n + 1}{(n - 1) (n + 8)} \cdot \frac{3}{3} + \frac{7}{3}$

Step 3

To write $\frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{(n - 1) (n + 8)}{(n - 1) (n + 8)}$ .

$\frac{n + 1}{(n - 1) (n + 8)} \cdot \frac{3}{3} + \frac{7}{3} \cdot \frac{(n - 1) (n + 8)}{(n - 1) (n + 8)}$

Step 4

Write each expression with a common denominator of $(n - 1) (n + 8) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{n + 1}{(n - 1) (n + 8)}$ by $\frac{3}{3}$ .

$\frac{(n + 1) \cdot 3}{(n - 1) (n + 8) \cdot 3} + \frac{7}{3} \cdot \frac{(n - 1) (n + 8)}{(n - 1) (n + 8)}$

Step 4.2

Multiply $\frac{7}{3}$ by $\frac{(n - 1) (n + 8)}{(n - 1) (n + 8)}$ .

$\frac{(n + 1) \cdot 3}{(n - 1) (n + 8) \cdot 3} + \frac{7 ((n - 1) (n + 8))}{3 ((n - 1) (n + 8))}$

Step 4.3

Reorder the factors of $(n - 1) (n + 8) \cdot 3$ .

$\frac{(n + 1) \cdot 3}{3 (n + 8) (n - 1)} + \frac{7 ((n - 1) (n + 8))}{3 ((n - 1) (n + 8))}$

Step 4.4

Reorder the factors of $3 ((n - 1) (n + 8))$ .

$\frac{(n + 1) \cdot 3}{3 (n + 8) (n - 1)} + \frac{7 ((n - 1) (n + 8))}{3 (n + 8) (n - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(n + 1) \cdot 3 + 7 ((n - 1) (n + 8))}{3 (n + 8) (n - 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{n \cdot 3 + 1 \cdot 3 + 7 (n - 1) (n + 8)}{3 (n + 8) (n - 1)}$

Step 6.2

Move $3$ to the left of $n$ .

$\frac{3 \cdot n + 1 \cdot 3 + 7 (n - 1) (n + 8)}{3 (n + 8) (n - 1)}$

Step 6.3

Multiply $3$ by $1$ .

$\frac{3 \cdot n + 3 + 7 (n - 1) (n + 8)}{3 (n + 8) (n - 1)}$

Step 6.4

Apply the distributive property.

$\frac{3 n + 3 + (7 n + 7 \cdot - 1) (n + 8)}{3 (n + 8) (n - 1)}$

Step 6.5

Multiply $7$ by $- 1$ .

$\frac{3 n + 3 + (7 n - 7) (n + 8)}{3 (n + 8) (n - 1)}$

Step 6.6

Expand $(7 n - 7) (n + 8)$ using the FOIL Method.

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Step 6.6.1

Apply the distributive property.

$\frac{3 n + 3 + 7 n (n + 8) - 7 (n + 8)}{3 (n + 8) (n - 1)}$

Step 6.6.2

Apply the distributive property.

$\frac{3 n + 3 + 7 n \cdot n + 7 n \cdot 8 - 7 (n + 8)}{3 (n + 8) (n - 1)}$

Step 6.6.3

Apply the distributive property.

$\frac{3 n + 3 + 7 n \cdot n + 7 n \cdot 8 - 7 n - 7 \cdot 8}{3 (n + 8) (n - 1)}$

Step 6.7

Simplify and combine like terms.

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Step 6.7.1

Simplify each term.

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Step 6.7.1.1

Multiply $n$ by $n$ by adding the exponents.

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Step 6.7.1.1.1

Move $n$ .

$\frac{3 n + 3 + 7 (n \cdot n) + 7 n \cdot 8 - 7 n - 7 \cdot 8}{3 (n + 8) (n - 1)}$

Step 6.7.1.1.2

Multiply $n$ by $n$ .

$\frac{3 n + 3 + 7 n^{2} + 7 n \cdot 8 - 7 n - 7 \cdot 8}{3 (n + 8) (n - 1)}$

Step 6.7.1.2

Multiply $8$ by $7$ .

$\frac{3 n + 3 + 7 n^{2} + 56 n - 7 n - 7 \cdot 8}{3 (n + 8) (n - 1)}$

Step 6.7.1.3

Multiply $- 7$ by $8$ .

$\frac{3 n + 3 + 7 n^{2} + 56 n - 7 n - 56}{3 (n + 8) (n - 1)}$

Step 6.7.2

Subtract $7 n$ from $56 n$ .

$\frac{3 n + 3 + 7 n^{2} + 49 n - 56}{3 (n + 8) (n - 1)}$

Step 6.8

Add $3 n$ and $49 n$ .

$\frac{52 n + 3 + 7 n^{2} - 56}{3 (n + 8) (n - 1)}$

Step 6.9

Subtract $56$ from $3$ .

$\frac{52 n + 7 n^{2} - 53}{3 (n + 8) (n - 1)}$

Step 6.10

Reorder terms.

$\frac{7 n^{2} + 52 n - 53}{3 (n + 8) (n - 1)}$