Problem 81 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{-9(-3)}{-6}\)

Step-by-Step Solution

Answer

The value of the expression is \(-4.5\).

Step 1: Understand the Expression

The expression given is a fraction \( \frac{-9(-3)}{-6} \). It involves multiplication in the numerator and a division operation.

Step 2: Compute the Numerator

Multiply the numbers in the numerator: \(-9\) and \(-3\). Know that the product of two negative numbers is positive. Therefore, \(-9 \times -3 = 27\).

Step 3: Write the Simplified Fraction

After computing the numerator, the expression becomes \( \frac{27}{-6} \).

Step 4: Divide the Numbers

Now, divide 27 by \(-6\). The division of a positive number by a negative number yields a negative result. \( \frac{27}{-6} = -4.5 \).

Step 5: Simplify if Necessary

Check if the result \(-4.5\) can be simplified further. In this case, \(-4.5\) is already in the simplest form.

Key Concepts

Multiplication of Integers

When multiplying integers, especially negative ones, it's important to remember a few key rules:

• The product of two negative numbers is positive. For example, \(-3 \times -3 = 9\).
• The product of a positive and a negative number is negative. For instance, \(3 \times -3 = -9\).
• The product of two positive numbers is positive, just as in standard multiplication.

In our exercise, the numbers \(-9\) and \(-3\) are both negative. Following the rule for the multiplication of two negatives, the product is positive. So, \(-9 \times -3\) results in \(27\). Understanding these rules helps to simplify complex expressions and handle integers accurately.

Division of Integers

Integer division can be a bit tricky, especially when negatives come into play. Here are some easy rules to guide you:

• Dividing two positive numbers, or two negative numbers, results in a positive quotient. For example, \(-10 \div -5 = 2\).
• Dividing a positive number by a negative number, or vice versa, gives a negative quotient. For example, \(20 \div -5 = -4\).

In the given exercise, the numerator \(27\) (positive) is divided by the denominator \(-6\) (negative). Applying the rule, a positive divided by a negative, yields a negative result: \(\frac{27}{-6} = -4.5\). This understanding makes it easier to handle equations involving large numbers and diverse signs.

Simplification of Fractions

Simplifying fractions is a useful skill in algebra that makes arithmetic operations and comparisons clearer. Here's how it works:

• First, identify the greatest common divisor (GCD) of the numerator and the denominator. Use this to divide both the numerator and the denominator, simplifying the fraction to its lowest terms.
• In cases where the fraction results in a decimal or whole number, such as \(-4.5\) in our example, check if it's already presented in the simplest form.
• If possible, simplify further until you cannot find any common divisors other than 1.

For the fraction \(\frac{27}{-6}\), the division gives \(-4.5\), which is already simplified. It's crucial to be comfortable with rewriting fractions in simpler forms to ensure accuracy in calculations and comparisons.