Problem 58 - Introductory Algebra for College Students

Question

Perform the indicated division or state that the expression is undefined. $$-80 \div 8$$

Step-by-Step Solution

Answer

The answer is -10, as \(-80 \div 8 = -10\).

Step 1: Identify the Divisor

Firstly, the divisor must be identified, which in this case is 8. Since this is not zero, the expression is defined.

Step 2: Perform the Division

The next step is to carry out the division: \(-80 \div 8\).

Key Concepts

Performing Division

Division in Algebra, akin to basic arithmetic, involves dividing a number (the dividend) by another number (the divisor). One way to handle this operation is by considering division as the process of finding out how many times the divisor fits into the dividend.

For example, when given the expression \( -80 \div 8 \), our goal is to determine how many times 8 is contained within -80. It should be noted that when dividing by a positive number, if our dividend is negative, the result will also be negative. So, \( -80 \div 8 \) results in \( -10 \).

In algebra, performing division is crucial for simplifying expressions and solving equations. It's important to execute this operation correctly to achieve an accurate solution.

Undefined Mathematical Expressions

Not all mathematical expressions have a defined value. In the realm of division, an expression becomes undefined when the divisor is zero. This isn't simply a rule; it reflects a deeper concept in mathematics concerning the properties of division.

An undefined expression in division, such as \( \frac{a}{0} \) (where 'a' is any real number), poses a problem because division is the inverse of multiplication and there is no number that you can multiply by zero to get 'a' (unless 'a' is also zero, but \( \frac{0}{0} \) is also undefined for other reasons).

Always be cautious: an undefined divisor is a critical pitfall in algebra that can invalidate an otherwise well-solved problem. It is essential to recognize and avoid division by zero to ensure mathematical accuracy.

Identifying the Divisor

Determining the divisor is a fundamental skill in division. It is defined as the number by which the dividend is to be divided. Recognizing the divisor correctly is paramount; since if it's mistaken for zero, the division operation becomes undefined, leading to an erroneous interpretation of the problem at hand.

In the example of \( -80 \div 8 \), the divisor is '8’. It is crucial to differentiate it from the dividend, which in this case is '-80'. Unlike the dividend, which can be zero or any real number, the divisor cannot be zero. This careful identification guides the direction of the division operation and helps in avoiding the creation of mathematically unsound expressions.