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Chapter 1: Problem 92

Evaluate each expression without a calculator Use a calculator to check. $$[6+2(-4)]^{2}$$w

Short Answer

Expert verified
4

Step by step solution

01

Simplify the expression inside the parentheses

First, solve the expression within the parentheses: 2(-4) = -8
02

Add the result to the number outside the parentheses

Next, add 6 to the result from the previous step: 6 + (-8) = -2
03

Square the result

Finally, square the result of the previous step: (-2)^2 = 4

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Key Concepts

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

Order of Operations
Understanding the order of operations is crucial when simplifying algebraic expressions. The order of operations tells us the sequence in which we should perform arithmetic operations to obtain a correct result. The commonly used acronym for this is PEMDAS:
  • P: Parentheses
  • E: Exponents (or powers, like squaring a number)
  • MD: Multiplication and Division (from left to right)
  • AS: Addition and Subtraction (from left to right)
Let's break it down in the context of the given problem: \[ [6+2(-4)]^{2} \] First, we solve the expression in the parentheses. Then, we take care of the multiplication inside the parentheses. Next, we perform any addition or subtraction outside the parentheses. Finally, we handle any exponents by squaring the number. Using PEMDAS ensures we simplify the expression in the correct sequence.
Parentheses Operations
Parentheses indicate which operations to perform first. Let's look at the part of the problem involving parentheses:
\[6+2(-4) \] First, evaluate the multiplication inside the parentheses.
Replace \(2(-4)\) with \(-8\):
\[6 + (-8)\text{ } \] Next, handle the addition or subtraction outside the parentheses. Since adding a negative number is effectively the same as subtracting:
\[6 - 8 = -2 \] Now, the expression inside the parentheses, \(6 + 2(-4)\), has been simplified to \(-2\). Handling parentheses correctly ensures we perform calculations in the intended sequence.
Squaring Numbers
Squaring a number means multiplying the number by itself. In our problem, once we have simplified the parentheses, we need to square the result: \[(-2)^{2} \] Squaring \(-2\) means multiplying \(-2\) by \(-2\): \(-2 \times -2 = 4\). Remember that multiplying two negative numbers results in a positive number. Squaring helps us to find the correct value, and it's an essential step, particularly when working through polynomial and quadratic expressions.
Basic Arithmetic
Basic arithmetic forms the foundation of all math problems. It includes:
  • Addition
  • Subtraction
  • Multiplication
  • Division
Let's see how it applies in the problem step-by-step:
1. Start with multiplication: \2(-4) = -8.\ This step applies the multiplication rule.
2. Follow with addition/subtraction:
\(6 + (-8) = -2\). Subtracting 8 from 6 gets us \(-2\).
3. Finally, use exponent rules: \((-2)^{2} = 4\).
Each of these operations builds on the previous one, showing how basic arithmetic principles are applied in combination to solve an algebraic expression.

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Most popular questions from this chapter

Simplify. $$ \frac{1}{6}-\frac{1}{3}\left(-6 x^{2} y-\frac{1}{2}\right) $$

Simplify each expression. $$ (-3 x+4)-(6 x-6) $$

Cooperative learning Work in a small group to make a list of the real numbers of the form \(\sqrt{n},\) where \(n\) is a natural number between 1 and 100 inclusive. Decide on a method for determining which of these numbers are rational and find them. Compare your group's method and results with other groups' work.

Find each product or quotient. $$ \frac{-6 x-9}{-3} $$

Simplify. $$ 3 w^{3}+5 w^{3}-4 w^{3}+12 w^{3}-2 w^{2} $$
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