Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(\frac{1}{h+6}(h+6)=1, \quad h \neq-6\)

Step-by-Step Solution

Answer

The statement illustrates the Cancellation Law in algebra.

Step 1: Identify the rule

The given equation is \(\frac{1}{h+6} \times (h+6) = 1\). The rule used here appears to be the cancellation law in algebra. The next step is to verify this law.

Step 2: Apply the rule

Apply the Cancellation Law in this expression. According to this law, a term in the numerator and a term in the denominator can be cancelled if they are identical and the denominator is not zero. Here, \((h+6)\) in the numerator and \((h+6)\) in the denominator are identical non-zero terms, so they cancel each other out to yield 1.

Step 3: Verify the non-zero denominator condition

\((h+6)\) in the denominator should not be zero, which means \(h\) should not be -6. This condition is given in the question, which confirms that the statement illustrates the cancellation law.

Key Concepts

Rules of Algebra

Algebra is built upon various key rules or properties that help in simplifying expressions and solving equations. Some of the fundamental rules include the associative, commutative, and distributive laws. In this exercise, the specific rule we're addressing is the

• Cancellation Law: This rule states that if a term appears both in the numerator and the denominator, and the expression in the denominator isn't zero, that term can be cancelled. For example, in the expression \( \frac{1}{h+6}(h+6)=1 \), both \( h+6 \) terms appear as a factor in the numerator and the denominator, enabling their cancellation.

These rules are foundational as they allow us to manipulate and solve algebraic expressions efficiently. When applying these rules, always be cautious about conditions such as zero denominators, which could invalidate the cancellations.

Numerator and Denominator

In a fraction, understanding the roles of the numerator and denominator is crucial. A fraction consists of two parts:

• Numerator: This is the top part of the fraction and represents how many parts of the whole are being considered.
• Denominator: This is the bottom part of the fraction and indicates the total number of equal parts the whole is divided into.

In the expression \( \frac{1}{h+6}(h+6) \), the term \( h+6 \) is both part of the numerator and the denominator. It's essential for evaluating the expression correctly and determining if the Cancellation Law can be applied. The cancelation process shows that these expressions in both parts of the fraction are identical, simplifying to 1.

Zero Denominator Condition

When working with fractions, one crucial condition to be vigilant about is the zero denominator condition. The denominator in a fraction can never be zero because division by zero is undefined and makes the expression invalid. Therefore, when examining a fraction like \( \frac{1}{h+6} \), we must ensure:

• The expression \( h+6 \) cannot be zero.
• This implies that \( h eq -6 \), which is explicitly stated in the problem.

Breaking this condition can result in unsolvable expressions or undefined situations. Hence, always check the expression set in the denominator and ensure it does not equal zero. This condition is critical when using the Cancellation Law to ensure the simplification is mathematically sound.