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Chapter 3: Problem 502

In the following exercises, evaluate. $$ (-2 r)^{2} \text { when } r=3 $$

Short Answer

Expert verified
36

Step by step solution

01

Identify the given variable

Here, the variable given is \( r \) and its value is 3.
02

Substitute the value of the variable

Substitute \( r = 3 \) into the expression \( (-2r)^{2} \). This gives us \( (-2 \times 3)^{2} \).
03

Simplify inside the parentheses

Calculate \( -2 \times 3 \), which equals \( -6 \). So the expression becomes \( (-6)^{2} \).
04

Apply the exponent

Square \( -6 \) by calculating \( (-6) \times (-6) \), which equals 36.
05

State the final answer

The final value of the expression \( (-2r)^{2} \) when \( r = 3 \) is 36.

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Key Concepts

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

Variable Substitution
We start by understanding what variable substitution means.
A variable is a symbol that represents a number. In algebra, letters like \( r \) or \( x \) are commonly used as variables.
Substitution involves replacing a variable with its given value.

In our exercise, the variable is \( r \) and its value is 3.
The given expression is \( (-2r)^{2} \).
For substitution, we will replace \( r \) with 3.
So, the expression becomes \( (-2 \times 3)^{2} \).
This is an essential first step in evaluating expressions because, without it, we cannot proceed to calculate accurately.
Simplifying Expressions
After substituting the variable, the next key concept is simplifying the expression.
Simplification often involves a series of algebraic operations done systematically.

Let's look at our substituted expression: \( (-2 \times 3)^{2} \).
To simplify this, we perform the multiplication inside the parentheses first.
  • Calculate \( -2 \times 3 \) which gives \( -6 \).

  • So, the expression simplifies to \( (-6)^{2} \).

Performing the operations within parentheses first follows the rules of PEMDAS/BODMAS, ensuring that the expression is correctly simplified before moving on to other operations.
Exponential Operations
The final part of our exercise involves understanding exponential operations.
Exponents tell us how many times to multiply a number by itself.

Here, our simplified expression is \( (-6)^{2} \).
We need to square \( -6 \), which means multiplying \( -6 \) by itself.
Hence, compute \( (-6) \times (-6) \).
A negative number squared always results in a positive number because two negative signs cancel out.
Therefore, \( (-6)^{2} = 36 \).

Exponential operations require precision, as the order of operations and sign rules are crucial to attaining the correct results.

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