Problem 46 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ (a-6)^{3} \cdot \frac{(a+2)^{2}}{a-6} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $(a-6)^{3} \cdot \frac{(a+2)^{2}}{a-6}$ Answer: $(a-6)^{2} \cdot (a+2)^{2}$

Step 1: Write down the expression

We have the expression: $$ (a-6)^{3} \cdot \frac{(a+2)^{2}}{a-6} $$

Step 2: Observe the common term in the numerator and the denominator

We can see that there is a common term (a-6) in both the numerator and the denominator.

Step 3: Simplify the expression by canceling out the common term

We can cancel out the common term (a-6) from the numerator and the denominator as follows: $$ (a-6)^{3} \cdot \frac{(a+2)^{2}}{a-6} = (a-6)^{2} \cdot (a+2)^{2} $$

Step 4: Write the final expression

The simplified expression after performing the multiplications and divisions is: $$ (a-6)^{2} \cdot (a+2)^{2} $$

Key Concepts

Polynomial Operations

Polynomial operations involve manipulating expressions that contain multiple terms added or subtracted together. This could mean multiplying, dividing, adding, or subtracting polynomials. In the given exercise, we're primarily focused on multiplication and division of polynomials. Let's delve into some basics to understand this better:

• **Multiplication**: When you multiply polynomials, you distribute each term in the first polynomial to each term in the second polynomial. This is often called the distributive property.
• **Division**: When dividing, particularly when it involves polynomials, you need to factor and cancel similar terms when possible. This helps in simplifying the expressions easily.

In this exercise, we simply multiplied the polynomials by distributing appropriately and then performed division using factor cancellation. This simplified our process greatly and allowed us to reach the final expression efficiently.

Exponent Rules

Exponent rules are fundamental in solving polynomial expressions like the one in this exercise. Mastering these rules helps simplify expressions involving powers more efficiently.

• **Power of a Power Rule**: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• **Product of Powers Rule**:When multiplying like bases, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• **Quotient of Powers Rule**:When dividing like bases, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In our problem, these rules assist in simplifying expressions like \((a-6)^3\) and understanding the multiplication of \((a-6)^2 \cdot (a+2)^2\). By adhering to these rules, we make simplification of expressions both efficient and accurate.

Factor Cancellation

Factor cancellation is a simplification strategy used particularly in fraction-like expressions. It involves identifying and removing common factors from the numerator and denominator. In algebra, recognizing factors and canceling them appropriately can greatly simplify expressions.

• **Identify Common Factors**:Identify elements that are present in both the numerator and the denominator. In our exercise, the factor \((a-6)\) is present both in the numerator and the denominator.
• **Cancel out**:Once common factors are identified, effectively remove them from both the top and bottom of the fraction. This process is akin to reducing fractions in arithmetic.

This technique simplifies the expression to \((a-6)^2 \cdot (a+2)^2\) by eliminating excess factors, showcasing the utility of factor cancellation in achieving a more manageable form.