Simplify ((a^3-27)/(a^2-9))/((a^2+3a+9)/(a+3))
The problem presents an algebraic expression that needs to be simplified. It involves a set of polynomials in the variable 'a' that are arranged in a complex fraction where one fraction is divided by another. Specifically, the expression is composed of a cubed-term polynomial (a^3) being subtracted by a constant (27) in the numerator of the first fraction and a squared-term polynomial (a^2) being subtracted by a constant (9) in the denominator of that same fraction. In the second fraction, the numerator is a quadratic polynomial (a^2 with terms involving a), and its denominator is a linear term (a+3). The task is to perform the necessary algebraic manipulations such as factoring, canceling common terms, and simplifying to reduce the expression to its simplest form.
$\frac{\frac{a^{3} - 27}{a^{2} - 9}}{\frac{a^{2} + 3 a + 9}{a + 3}}$
Answer
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:
Reciprocal Multiplication: When dividing by a fraction, multiply by its reciprocal.
Difference of Cubes: The formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ is used to factor cubic differences.
Difference of Squares: The formula $a^2 - b^2 = (a + b)(a - b)$ is used to factor quadratic differences.
Factoring: Recognizing and factoring perfect square trinomials and cubic expressions.
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.
