Problem 85 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{-9(-3)}{-6} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{9}{2}\).

Step 1: Simplify the Numerator

The first part of this expression is the operation in the numerator, \(-9 \times (-3)\).A negative number times a negative number results in a positive product. Therefore: \(-9 \times (-3) = 27\). So the expression simplifies to \(\frac{27}{-6}\).

Step 2: Simplify by Division

Now we simplify the expression \(\frac{27}{-6}\) by dividing the numerator by the denominator. When dividing a positive number by a negative number, the result is negative. So, \(27 \div -6 = -4.5\).

Step 3: Final Simplification

Since dividing resulted in a decimal, it's often preferable to express this as a fraction in simplest form. We begin by recognizing \(27\) and \(6\) share a factor of \(3\). So, \(\frac{27}{6} = \frac{9}{2}\) because dividing both \(27\) and \(6\) by \(3\) gives these values.

Step 4: Apply the Negative Sign

Remember our earlier step where we noted dividing by a negative makes the result negative. Therefore, the simplified expression is:\(-\frac{9}{2}\).

Key Concepts

Negative Numbers

Understanding negative numbers is crucial in algebra simplification. Negative numbers are those less than zero and are indicated by a minus sign (-). When you multiply two negative numbers, the result is a positive number because the two negatives cancel each other out. This is a fundamental rule in mathematics to remember:

• If you multiply a negative by a positive, the product is negative.
• If you multiply two negatives, the product is positive.

In our exercise, we multiply \(-9\times (-3)\). Since both numbers are negative, the product is positive: \(9 \times 3 = 27\). This makes the expression easier to handle and sets the stage for further simplification.
Recognizing how negative numbers interact in multiplication is a vital skill. This understanding will assist you in navigating more complex algebraic expressions.

Numerator and Denominator

The numerator and denominator are essential parts of any fraction. The numerator is the top number, showing how many parts we have, while the denominator is the bottom number, indicating how many parts make up a whole. In simplifying fractions, observe these components clearly:

• The expression begins with a numerator: \(-9(-3)\), which simplifies to 27.
• The denominator is \(-6\).

It's key to execute mathematical operations such as multiplication or division on these individual parts first before putting them together.
In our problem, once the operation in the numerator is simplified to 27, we're left with \(\frac{27}{-6}\). You can now proceed to evaluate and simplify the whole fraction. A solid grasp of these concepts assists in breaking down fractions efficiently and correctly.

Fraction Simplification

Simplifying fractions is converting them to their simplest form. This makes the fraction easier to understand and use. Fractions are in simplest form when the greatest common factor (GCF) of the numerator and the denominator is one. Here is how it was done:

• Recognize the GCF of the numerator and denominator. In this case, 27 and 6 share a factor of 3.
• Divide both the numerator and the denominator by 3. This gives: \(\frac{27}{6} = \frac{9}{2}\).

Although this yielded \(\frac{9}{2}\) as a positive, you then apply the negative sign from dividing:\(-\frac{9}{2}\).
Having a positive result from simplification initially might seem odd, but remembering how the division sign affects the whole outcome is key to correct fraction simplification. By maintaining clarity on how negative signs influence results, you ensure your simplified fractions are accurate and reflective of the operations performed.