Problem 31 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$(-2)^{3}-3^{2}$$

Step-by-Step Solution

Answer

The simplified expression is -17.

Step 1: Evaluate the Exponentiation of the First Term

The first term in the expression is \((-2)^{3}\). The base is -2, and the exponent is 3, which means we are multiplying -2 by itself two more times: \((-2) \times (-2) \times (-2)\). Work through the multiplication: 1. Multiply the first two factors: \((-2) \times (-2) = 4\).2. Multiply the result by -2: \(4 \times (-2) = -8\).Thus, \((-2)^{3} = -8\).

Step 2: Evaluate the Exponentiation of the Second Term

The second term is \(3^{2}\). The base is 3, and the exponent is 2, which means we are multiplying 3 by itself: \(3 \times 3 = 9\).Thus, \(3^{2} = 9\).

Step 3: Subtract the Second Term from the First Term

Now that we have both parts of the expression simplified, we can subtract the second term from the first. Substitute the simplified values into the expression:- First term: \(-8\)- Second term: \(9\)Calculate the subtraction: \(-8 - 9 = -17\).Thus, the simplified expression is \(-17\).

Key Concepts

Exponentiation

Exponentiation allows us to work with numbers raised to a power. When we see a term like \((-2)^3\), we identify that -2 is the base and 3 is the exponent. The exponent indicates how many times we multiply the base by itself.

Let's break this down:

• Base: -2, in our example, needs to be multiplied by itself.
• Exponent: 3 tells us to perform the multiplication three times.

Performing \((-2) \times (-2)\) gives us 4, because multiplying two negative numbers produces a positive result. Then, multiplying the positive 4 with another -2 results in -8, hence \((-2)^3 = -8\).

This range of multiplication shows how exponentiation is a shortcut for repeated multiplication and it can make calculations more efficient and straightforward.

Negative Numbers

Working with negative numbers involves understanding several basic rules. Negative numbers have values less than zero, and they follow specific rules when used in calculations.

Here are some key points:

• Multiplying two negative numbers results in a positive number.
• Multiplying a positive number with a negative number gives a negative result.
• Subtraction, when involving negative numbers, can sometimes result in an increase. For instance, subtracting a negative number is equivalent to adding a positive number.

In the exercise \((-2)^3\), we multiply -2 by itself, which involves the rules above. The first step is \((-2) \times (-2) = 4\), and next we compute \(4 \times (-2) = -8\).

Recognizing these rules helps simplify complex expressions and avoids confusion, particularly in exercises involving exponentiation and subtraction.

Simplification of Expressions

Simplifying expressions is the process of reducing a mathematical expression to its simplest form. This typically involves performing operations and reducing expressions to a single numerical result or a simpler form.

The main steps in simplification include:

• Evaluating exponentiation: Simplify terms with exponents first, as seen in \((-2)^3\) and \(3^2\).
• Following arithmetic order: After dealing with exponents, proceed with other operations such as multiplication, division, addition, or subtraction.
• Combining like terms: When combining, ensure to follow rules relating to negative values and arithmetic operations.

In the original exercise, once we evaluate both exponents, we end up with -8 and 9 as separate expressions to be combined through subtraction.

By following a step-by-step approach, simplification becomes manageable even when dealing with complex numerical expressions.

Subtraction in Algebra

Subtraction in algebra involves removing a quantity from another. It is represented by the minus sign \(-\), and in many cases, subtraction with negative numbers can seem tricky.

Below is how you can handle subtraction efficiently:

• When subtracting a larger positive number from a smaller one, the result will be a negative number.
• Subtracting negative numbers is akin to adding their positive counterparts. For example, \(-8 - 9 = -17\) because you treat the operation as -8 plus the negative of 9.

In our example, once we determined \((-2)^3 = -8\) and \(3^2 = 9\), the subtraction part was straightforward, as you simply subtract the two simplified values.

By learning how to deal with subtraction, especially involving negatives, you enhance not only your numerical accuracy but also your understanding of more complex algebraic expressions.