Algebra Examples

$\frac{n^{2} + 12 n + 27}{3 n^{3} + 9 n^{2}} \div \frac{2 n^{2} + 18 n}{B} = 1$

Step 1

Factor each term.

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Step 1.1

Factor $n^{2} + 12 n + 27$ using the AC method.

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Step 1.1.1

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 $27$ and whose sum is $12$ .

$3, 9$

Step 1.1.2

Write the factored form using these integers.

$\frac{(n + 3) (n + 9)}{3 n^{3} + 9 n^{2}} \div \frac{2 n^{2} + 18 n}{B} = 1$

Step 1.2

Factor $3 n^{2}$ out of $3 n^{3} + 9 n^{2}$ .

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Step 1.2.1

Factor $3 n^{2}$ out of $3 n^{3}$ .

$\frac{(n + 3) (n + 9)}{3 n^{2} (n) + 9 n^{2}} \div \frac{2 n^{2} + 18 n}{B} = 1$

Step 1.2.2

Factor $3 n^{2}$ out of $9 n^{2}$ .

$\frac{(n + 3) (n + 9)}{3 n^{2} (n) + 3 n^{2} (3)} \div \frac{2 n^{2} + 18 n}{B} = 1$

Step 1.2.3

Factor $3 n^{2}$ out of $3 n^{2} (n) + 3 n^{2} (3)$ .

$\frac{(n + 3) (n + 9)}{3 n^{2} (n + 3)} \div \frac{2 n^{2} + 18 n}{B} = 1$

Step 1.3

Reduce the expression $\frac{(n + 3) (n + 9)}{3 n^{2} (n + 3)}$ by cancelling the common factors.

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Step 1.3.1

Cancel the common factor.

$\frac{(n + 3) (n + 9)}{3 n^{2} (n + 3)} \div \frac{2 n^{2} + 18 n}{B} = 1$

Step 1.3.2

Rewrite the expression.

$\frac{n + 9}{3 n^{2}} \div \frac{2 n^{2} + 18 n}{B} = 1$

Step 1.4

Factor $2 n$ out of $2 n^{2} + 18 n$ .

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Step 1.4.1

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

$\frac{n + 9}{3 n^{2}} \div \frac{2 n (n) + 18 n}{B} = 1$

Step 1.4.2

Factor $2 n$ out of $18 n$ .

$\frac{n + 9}{3 n^{2}} \div \frac{2 n (n) + 2 n (9)}{B} = 1$

Step 1.4.3

Factor $2 n$ out of $2 n (n) + 2 n (9)$ .

$\frac{n + 9}{3 n^{2}} \div \frac{2 n (n + 9)}{B} = 1$

Step 1.5

To divide by a fraction, multiply by its reciprocal.

$\frac{n + 9}{3 n^{2}} \cdot \frac{B}{2 n (n + 9)} = 1$

Step 1.6

Cancel the common factor of $n + 9$ .

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Step 1.6.1

Factor $n + 9$ out of $2 n (n + 9)$ .

$\frac{n + 9}{3 n^{2}} \cdot \frac{B}{(n + 9) (2 n)} = 1$

Step 1.6.2

Cancel the common factor.

$\frac{n + 9}{3 n^{2}} \cdot \frac{B}{(n + 9) (2 n)} = 1$

Step 1.6.3

Rewrite the expression.

$\frac{1}{3 n^{2}} \cdot \frac{B}{2 n} = 1$

Step 1.7

Multiply $\frac{1}{3 n^{2}}$ by $\frac{B}{2 n}$ .

$\frac{B}{3 n^{2} (2 n)} = 1$

Step 1.8

Multiply $2$ by $3$ .

$\frac{B}{6 n^{2} n} = 1$

Step 1.9

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

$\frac{B}{6 (n^{1} n^{2})} = 1$

Step 1.10

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

$\frac{B}{6 n^{1 + 2}} = 1$

Step 1.11

Add $1$ and $2$ .

$\frac{B}{6 n^{3}} = 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$6 n^{3}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$6 n^{3}$

Step 3

Multiply each term in $\frac{B}{6 n^{3}} = 1$ by $6 n^{3}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{B}{6 n^{3}} = 1$ by $6 n^{3}$ .

$\frac{B}{6 n^{3}} (6 n^{3}) = 1 (6 n^{3})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$6 \frac{B}{6 n^{3}} n^{3} = 1 (6 n^{3})$

Step 3.2.2

Cancel the common factor of $6$ .

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Step 3.2.2.1

Factor $6$ out of $6 n^{3}$ .

$6 \frac{B}{6 (n^{3})} n^{3} = 1 (6 n^{3})$

Step 3.2.2.2

Cancel the common factor.

$6 \frac{B}{6 n^{3}} n^{3} = 1 (6 n^{3})$

Step 3.2.2.3

Rewrite the expression.

$\frac{B}{n^{3}} n^{3} = 1 (6 n^{3})$

Step 3.2.3

Cancel the common factor of $n^{3}$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{B}{n^{3}} n^{3} = 1 (6 n^{3})$

Step 3.2.3.2

Rewrite the expression.

$B = 1 (6 n^{3})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $6 n^{3}$ by $1$ .

$B = 6 n^{3}$