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Chapter 1: Problem 61

Evaluate. If an expression is undefined, say so. $$\frac{4+2 \cdot 7}{3 \cdot 2-9}$$

Short Answer

Expert verified
-6

Step by step solution

01

- Apply Order of Operations (PEMDAS)

Start by evaluating the numerator and the denominator separately, applying the order of operations (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
02

- Simplify the Numerator

In the numerator: First, perform the multiplication: \( 2 \times 7 = 14 \) Next, perform the addition: \( 4 + 14 = 18 \)So, the numerator is 18.
03

- Simplify the Denominator

In the denominator:First, perform the multiplication: \( 3 \times 2 = 6 \) Next, perform the subtraction: \( 6 - 9 = -3 \)So, the denominator is -3.
04

- Combine the Results

Now, substitute the simplified numerator and denominator back into the fraction:\( \frac{18}{-3} = -6 \)So, the expression evaluates to -6.

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Key Concepts

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

PEMDAS
To solve any mathematical expression correctly and confidently, we need to follow a specific order or set of rules known as PEMDAS.
PEMDAS stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

By following these rules, we can avoid mistakes and inconsistencies in our calculations. Whenever we see a mathematical expression, we apply PEMDAS by looking for operations inside parentheses first, then evaluating exponents, followed by handling multiplication and division, and finally performing any addition and subtraction.
Numerator Simplification
When simplifying a fraction, start by focusing on the numerator鈥攖he top part of the fraction. Let's take the given example:
\(\frac{4+2 \times 7}{3 \times 2-9}\).
First, apply PEMDAS to the numerator:
  • Identify multiplication: \(2 \times 7 = 14\)

Now, add the result to the remaining number in the numerator:
  • \( 4 + 14 = 18 \)

Thus, the numerator simplifies to 18.
Denominator Simplification
Next, we simplify the denominator鈥攖he bottom part of the fraction. Applying PEMDAS to the denominator of our example, \(3 \times 2-9\), we:
  • Identify multiplication: \(3 \times 2 = 6\)

Followed by performing the subtraction:
  • \(6 - 9 = -3\)

Thus, the denominator simplifies to -3.
Evaluating Expressions
Once both the numerator and denominator are simplified, we can evaluate the expression by dividing the simplified numerator by the simplified denominator. For the fraction:
\(\frac{18}{-3}\),
we simply divide 18 by -3:
  • \( \frac{18}{-3} = -6 \)

Hence, the complete expression evaluates to -6. Always simplify and resolve each part of the expression step by step using PEMDAS to ensure accuracy.

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Most popular questions from this chapter

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