Problem 13 Let $a=-2$ and $b=3 .$ Evalu... [FREE SOLUTION]

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Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 13

Let $a=-2$ and $b=3 .$ Evaluate each expression. $$-\left|a^{2}-b^{2}\right|$$

Short Answer

Expert verified

The value of the expression $-|a^{2}-b^{2}|$ is -5.

Step by step solution

Substitute the given values

Start by substituting the given values for $a$ and $b$ into $a^{2}-b^{2}$. So it becomes $(-2)^{2}-3^{2} = 4-9 = -5$.

Apply the Absolute Value

After calculating the difference, we apply the absolute value to it, which changes the negative result into positive. So we get $|-5| = 5$

Apply the Negative Sign

Finally, the negative sign in front of the absolute value operation must be applied. So, we get $-|a^{2}-b^{2}| = -5$.

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Key Concepts

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

Substitution in Algebraic Expressions

Understanding how to substitute values into algebraic expressions is a fundamental skill in mathematics. It can simplify expressions and help in problem-solving. When you have an algebraic expression like $ a^2 - b^2 $, we can replace the variables with actual values to evaluate it.
Let's break it down:

Identify the expression you need to evaluate. Here it's $ a^2 - b^2 $.
Find the given values for the variables. In this exercise, $ a = -2 $ and $ b = 3 $.
Substitute these values into the expression: Replace $ a $ with $ -2 $ and $ b $ with $ 3 $, so it becomes $ (-2)^2 - 3^2 $.
Calculate using these substitutions, leading to $ 4 - 9 $.

Substitution transforms a general algebraic expression into a numerical one, ready for straightforward calculation.

Exponents

Exponents are a critical mathematical concept which represent repeated multiplication of a number. For instance, $ a^2 $ means multiplying $ a $ by itself. Understanding how to work with exponents is crucial for evaluating expressions effectively.
In the exercise, we dealt with:

Finding $ (-2)^2 $. This means $ -2 \times -2 $. A negative number times itself results in a positive number, so $ (-2)^2 = 4 $.
Calculating $ 3^2 $, or $ 3 \times 3 $. This yields a positive 9, so $ 3^2 = 9 $.

By handling exponents correctly, we make sure that the base units multiply accurately, leading to the right results when further computation proceeds.

Negative Numbers

Negative numbers can be tricky, but understanding how to work with them is essential. They have unique properties compared to positive numbers, and can significantly affect the outcome of an expression.
In this exercise, negative numbers appear in a few ways:

Starting with the initial substitution, $ a = -2 $ affects the squared result due to its negative nature, yielding $ 4 $ instead of $ -4 $.
After performing the subtraction, $ 4 - 9 $ resulted in $ -5 $, showcasing how operations can produce negative results.
The absolute value step turns $ -5 $ into $ 5 $, neutralizing the negative sign.
Finally, applying a negative sign converts $ 5 $ back to $ -5 $.

By understanding these characteristics and operations involving negative numbers, we accurately determine the final result.

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91影视

Let \(a=-2\) and \(b=3 .\) Evaluate each expression. $$-\left|a^{2}-b^{2}\right|$$

Short Answer

Step by step solution

Substitute the given values

Apply the Absolute Value

Apply the Negative Sign

Key Concepts

Substitution in Algebraic Expressions

Exponents

Negative Numbers

One App. One Place for Learning.

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