Problem 24 - Prealgebra

Question

Use the definition of exponents to expand each of the following expressions. Then multiply according to the rule for multiplication. a. \((-4)^{3}\) b. \(-4^{3}\)

Step-by-Step Solution

Answer

Both expressions evaluate to \(-64\).

Step 1: Understanding Exponents

An exponent tells us how many times to multiply the base by itself. In the expression \( a^b \), \( a \) is the base and \( b \) is the exponent. Multiply the base \( a \) by itself \( b \) times.

Step 2: Expanding Expression a

For \((-4)^3\), treat \(-4\) as the whole base. Write the expression as multiplication of three \((-4)\): \[(-4) \times (-4) \times (-4)\].

Step 3: Multiplying Expression a

To calculate \((-4) \times (-4)\), we get \(16\). Then multiply \(16 \times (-4)\) to get \(-64\). So, \((-4)^3 = -64\).

Step 4: Expanding Expression b

For \(-4^3\), only \(4\) is the base and the negative sign is separate. Write the expression expanding just \(4\): \(-1 \times (4 \times 4 \times 4)\).

Step 5: Multiplying Expression b

Calculate \(4 \times 4\), which is \(16\). Then \(16 \times 4 = 64\). Multiply by \(-1\) to account for the negative sign: \(-64\). So, \(-4^3 = -64\).

Key Concepts

Base and Exponent

Exponents are a shorthand way to show repeated multiplication of the same number. In the expression \(a^b\), \(a\) is called the base and \(b\) is the exponent. This means you multiply the base by itself as many times as the exponent indicates. For example, in \((-4)^3\), \(-4\) is the base, and 3 is the exponent. Therefore, we write it as \((-4) \times (-4) \times (-4)\). Each time we see an exponent, it reminds us of how many identical bases we need to multiply together. This simplification helps in performing large computations more manageably. Make sure to distinguish clearly whether the negative sign should be included with the base or not, as this changes how you expand the expression. It’s important to recognize the distinction between expressions like \((-4)^3\) and \(-4^3\), as they result from different interpretations of the base.

Multiplication of Integers

When multiplying integers, there are a few simple rules to remember:

• The product of two positive integers is a positive integer.
• The product of two negative integers is a positive integer.
• The product of a positive integer and a negative integer is a negative integer.

Applying these rules makes it easier to handle expressions with multiple integer multiplications. For example, in the expression \((-4) \times (-4) \times (-4)\), we multiply the first two \((-4)\), which gives us a positive 16, because multiplying two negative numbers results in a positive. Then we multiply 16 by another \((-4)\), resulting in \(-64\), because the product of a positive and a negative is negative. Keeping these simple rules in mind helps avoid mistakes and ensures accurate calculations.

Negative Numbers and Exponents

Negative numbers and their handling with exponents can sometimes be confusing, but understanding the order of operations helps. When a negative number is raised to a power, like \((-4)^3\), and the base includes the negative sign within parentheses, it’s treated entirely as the base. In this case, since the exponent is odd, the result is negative because multiplying an odd number of negative bases results in a negative number.However, in an expression like \(-4^3\), the negative sign is not part of the base but rather applies to the final result. The exponent 3 only applies to the 4, not the negative sign. This means we calculate \(4^3\) first, which is 64, and then apply the negative sign, resulting in \(-64\).It's crucial to read the expressions carefully:

• If a negative number is within parentheses and an exponent follows immediately, treat the negative sign as part of the base.
• If a negative sign is outside the base and not included in parentheses, compute the exponent first and then apply the sign.

This understanding helps clarify common pitfalls and ensures correct evaluation of expressions involving negative numbers and exponents.