Problem 25 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ \left(-\frac{1}{3}\right)\left(\frac{2}{5}\right) $$

Step-by-Step Solution

Answer

The result of the operation is \(-\frac{2}{15}\).

Step 1: Identify the operation

The expression provided is \[\left(-\frac{1}{3}\right)\left(\frac{2}{5}\right)\]which is a multiplication of two fractions.

Step 2: Multiply the numerators

To multiply the two fractions, first multiply the numerators:\[-1 \times 2 = -2\]

Step 3: Multiply the denominators

Next, multiply the denominators of the fractions:\[3 \times 5 = 15\]

Step 4: Combine the results

Combine the results from Steps 2 and 3 to form the new fraction:\[\frac{-2}{15}\]

Step 5: Check for simplification

The fraction \(\frac{-2}{15}\) cannot be simplified further since 2 and 15 have no common factors other than 1.

Key Concepts

Real Numbers

Real numbers are the set of numbers encompassing all rational and irrational numbers. If we think about the number line, every point on it represents a real number. This includes numbers like fractions, whole numbers, and even decimals.
When performing operations with real numbers, it's important to consider whether the numbers involved are positive or negative. In the exercise, \(-\frac{1}{3}\) and \ \frac{2}{5}\ are both real numbers, and one is negative. This affects the result of the multiplication.

• Rational numbers: These can be expressed as the quotient of two integers, like \(\frac{1}{3}\).
• Irrational numbers: These cannot be expressed as a simple fraction, such as \(\sqrt{2}\).

The operation of multiplication results in another real number, a rule that always holds when multiplying together any two real numbers.

Numerators and Denominators

Every fraction is made up of a numerator and a denominator. These are the two numbers you see in a fraction: one on top and one on the bottom, separated by a line. The top number is known as the numerator, and the bottom number is the denominator.

• Numerator: Represents the number of parts we have.
• Denominator: Shows the total number of equal parts the whole is divided into.

In the exercise, when multiplying fractions \(-\frac{1}{3}\) and \(\frac{2}{5}\), it involves multiplying the numerators and the denominators separately. Multiply the numerators \(-1\) and \(2\) to get \(-2\). Similarly, multiply the denominators \(3\) and \(5\) to get \(15\).
This forms a new fraction: \(\frac{-2}{15}\). Fractions are essentially representations of division: the numerator divided by the denominator represents a part of a whole.

Fraction Simplification

Simplifying a fraction involves breaking it down into its simplest form. This means that the numerator and the denominator should be the smallest possible integers while keeping the value of the fraction the same.
To simplify, we look for any common factors that the numerator and the denominator might share, other than 1.

• If such common factors exist, divide them out.
• If no common factors are found, the fraction is already in its simplest form.

For example, in the fraction \(\frac{-2}{15}\), the numbers 2 and 15 do not have any common factors other than 1. Therefore, the fraction cannot be simplified any further. This process of checking is crucial whenever you multiply or add fractions, ensuring you're working with the simplest values possible.