Simplify ((a^3-27)/(a^2-9))/((a^2+3a+9)/(a+3))

Solution:

Step:1

Take the reciprocal of the denominator and multiply it with the numerator.

$$\frac{a^3-27}{a^2-9}\times \frac{a+3}{a^2+3a+9}$$

Step:2

Factor the numerator.

Step:2.1

Express $27$ as $3^3$.

$$\frac{a^3-3^3}{a^2-9}\times \frac{a+3}{a^2+3a+9}$$

Step:2.2

Apply the difference of cubes formula, $a^3-b^3=(a-b)(a^2+ab+b^2)$, where $a=a$ and $b=3$.

$$\frac{(a-3)(a^2+3a+9)}{a^2-9}\times \frac{a+3}{a^2+3a+9}$$

Step:2.3

Refactor.

Step:2.3.1

Reorder the terms.

$$\frac{(a-3)(a^2+3a+9)}{a^2-9}\times \frac{a+3}{a^2+3a+9}$$

Step:2.3.2

Simplify the expression.

$$\frac{(a-3)(a^2+3a+9)}{a^2-9}\times \frac{a+3}{a^2+3a+9}$$

Step:3

Factor the denominator.

Step:3.1

Express $9$ as $3^2$.

$$\frac{(a-3)(a^2+3a+9)}{a^2-3^2}\times \frac{a+3}{a^2+3a+9}$$

Step:3.2

Use the difference of squares formula, $a^2-b^2=(a+b)(a-b)$, where $a=a$ and $b=3$.

$$\frac{(a-3)(a^2+3a+9)}{(a+3)(a-3)}\times \frac{a+3}{a^2+3a+9}$$

Step:4

Eliminate common factors.

Step:4.1

Remove the common term $a^2+3a+9$.

Step:4.1.1

Extract $a^2+3a+9$ from the numerator.

$$\frac{(a^2+3a+9)(a-3)}{(a+3)(a-3)}\times \frac{a+3}{a^2+3a+9}$$

Step:4.1.2

Cancel out the common term.

$$\frac{\cancel{(a^2+3a+9)}(a-3)}{(a+3)(a-3)}\times \frac{a+3}{\cancel{(a^2+3a+9)}}$$

Step:4.1.3

Simplify the fraction.

$$\frac{a-3}{(a+3)(a-3)}(a+3)$$

Step:4.2

Cancel out the common term $a+3$.

Step:4.2.1

Eliminate the common factor.

$$\frac{a-3}{\cancel{(a+3)}(a-3)}\cancel{(a+3)}$$

Step:4.2.2

Condense the expression.

$$\frac{a-3}{a-3}$$

Step:4.3

Remove the common term $a-3$.

Step:4.3.1

Cancel the common factor.

$$\frac{\cancel{(a-3)}}{\cancel{(a-3)}}$$

Step:4.3.2

Finalize the simplification.

$$1$$

Knowledge Notes:

The problem-solving process involves simplifying a complex rational expression by factoring and canceling common terms. The key knowledge points include:

1. Reciprocal Multiplication: When dividing by a fraction, multiply by its reciprocal.

2. Difference of Cubes: The formula $a^3-b^3=(a-b)(a^2+ab+b^2)$ is used to factor cubic differences.

3. Difference of Squares: The formula $a^2-b^2=(a+b)(a-b)$ is used to factor quadratic differences.

4. Factoring: Recognizing and factoring perfect square trinomials and cubic expressions.

5. Cancellation: Simplifying expressions by canceling out common factors in the numerator and denominator.

Understanding these concepts is critical for simplifying rational expressions and solving algebraic equations.