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Chapter 4: Problem 43

Evaluate each expression. $$-3\left(6^{2}-2^{3}\right)$$

Short Answer

Expert verified
-84

Step by step solution

01

Simplify Inside the Brackets

Before applying the multiplication indicated by the negative outside the brackets, simplify the expression \(6^{2}-2^{3}\). Using exponentiation rules, this evaluates to \(36-8\).
02

Perform the Subtraction

Subtract 8 from 36 to get 28.
03

Apply the Multiplication

Now, apply the multiplication by -3 to our result from step 2. \( -3 \times 28 \) equals -84.

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Key Concepts

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

Exponentiation Rules
Exponentiation involves raising a number, known as a base, to a power by multiplying the base by itself a specific number of times indicated by the exponent. In algebra, understanding exponent rules simplifies complex expressions. Let's break down the rules applied in our problem:
  • Square Rule: The expression \(6^{2}\) means 6 multiplied by itself: \(6 \times 6 = 36\).
  • Cubic Rule: Similarly, \(2^{3}\) means 2 multiplied by itself three times: \(2 \times 2 \times 2 = 8\).
These rules make it easier to simplify arithmetic expressions. By applying them, we simplify \(6^{2}-2^{3}\) to \(36 - 8\) before proceeding with other operations.
Order of Operations
In mathematics, the order of operations is a set of rules that dictates the sequence in which different calculations are performed. This ensures that everyone interprets and solves equations the same way. The key rule to remember is PEMDAS:
  • P: Parentheses first
  • E: Exponents (i.e., powers and square roots, etc.)
  • M/D: Multiplication and Division (left to right)
  • A/S: Addition and Subtraction (left to right)
In our exercise, first, we simplify the expression inside the parentheses, \(6^{2}-2^{3}\), by handling the exponents. This follows the E in PEMDAS. Then, subtraction is performed, resulting in 28. Finally, the multiplication by \(-3\) is applied, showing the correct translation of the operations.
Negative Numbers
Negative numbers represent values less than zero and are often used in various mathematical operations. When working with these numbers, it's important to consider the following rules:
  • Multiplying a Positive by a Negative: The result is always negative. For example, multiplying 28 by \(-3\) gives \(-84\).
  • Arithmetic Operations: When adding or subtracting, treat negative numbers as values below zero; think of adding them as moving left on a number line from zero.
In our exercise, the multiplication of 28 by the negative \(-3\) highlights this concept, providing a result of \(-84\). Understanding how negative numbers interact with positive numbers, especially in multiplication, is crucial for solving algebraic expressions correctly.

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