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Chapter 3: Problem 75

Evaluate the given expression. $$\frac{-4(-2)(-6)}{-4(-2)-6}$$

Short Answer

Expert verified
\(-24\)

Step by step solution

01

- Simplify Numerator

First, evaluate the product in the numerator of the fraction. Multiply \(-4 \times -2 \times -6\).
02

- Solution

\[ -4 \times -2 = 8 \] \(+8 \times -6 = -48 \). Thus, the numerator simplifies to \(-48\).
03

- Simplify Denominator

Evaluate and simplify the expression in the denominator: \(-4 \times -2 - 6\). Calculate the product and then subtract 6.
04

- Solution

\[ -4 \times -2 = 8 \] and subtracting 6 gives us \(+8 - 6 = 2\). Thus, the denominator simplifies to \(2\).
05

- Divide Numerator by Denominator

Finally, divide the simplified numerator \(-48\) by the simplified denominator \(2\).
06

- Solution

\[ \frac{-48}{2} = -24 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Fractions
To simplify a fraction, you need both the numerator (the top part) and the denominator (the bottom part) in their simplest forms. This usually involves reducing both parts by their greatest common divisor (GCD).
  • For example, with the fraction \(\frac{8}{12}\), the GCD of 8 and 12 is 4.
  • So, dividing both by 4, we get \(\frac{2}{3}\).
In the exercise given, the fraction is \(\frac{-48}{2}\). No need to find a GCD since the denominator is already 2. Instead, we divide 48 by 2 straightaway to get -24. Simplifying means making the fraction as easy to understand as possible.
Multiplication of Integers
Multiplying integers is straightforward if you follow the signs:
  • Multiplying two positive numbers gives a positive result.
  • Multiplying two negative numbers also gives a positive result.
  • Multiplying one positive and one negative number gives a negative result.
In our example: \(-4 \times -2 = 8\) because both numbers are negative.
Then, multiplying this result by -6: \(+8 \times -6 = -48\). Remember the rule: Positive multiplied by Negative equals Negative. Always keep an eye on the signs.
Order of Operations
When solving algebraic expressions, the order of operations is crucial. It鈥檚 often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
1. **Parentheses**. Always start inside the parentheses.
2. **Exponents**. Then, solve any exponents.
3. **Multiplication and Division**. Next, perform all the multiplication and division, from left to right.
4. **Addition and Subtraction**. Lastly, do all the addition and subtraction, from left to right.

In our specific exercise, we followed PEMDAS:
  • We performed the multiplications in the numerator: \( -4 \times -2 \) first, then \( \times -6\) .
  • Next, we worked out the multiplication in the denominator: \(-4 \times -2 - 6\).

  • Finally, we did the division: \(\frac{-48}{2} = -24 \).
Following the order of operations ensures that we get the correct and consistent result every time.

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