Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. Write fractional answers in simplest form.\((-3)^{4}\)

Step-by-Step Solution

Answer

The expression \((-3)^{4}\) evaluates to 81.

Step 1: Identify the Base and Exponent

In the expression \((-3)^{4}\), -3 is the base and 4 is the exponent. According to the rule of exponents, any number (negative or positive) raised to an even power will be a positive number. So the result of this calculation will be positive.

Step 2: Calculate the Expression

Multiply -3 by itself 4 times: \[(-3) \times (-3) \times (-3) \times (-3) = 81 \]. The result of \((-3)^{4}\) is 81.

Key Concepts

Base and Exponent

When you look at an expression like \((-3)^4\), it's important to identify the base and the exponent.
The base in this expression is \(-3\), which is the number you will multiply by itself.
The exponent, in this case, is \(4\), which tells you how many times you should multiply the base. Here's a quick checklist to figure out base and exponent:

• The base is the larger number or term that appears below or before the exponent.
• The exponent, a smaller number, usually appears as a superscript to the right of the base.

The base can be any number, negative or positive, and the exponent tells us how many times to use the base in multiplication.

Rule of Exponents

Exponentiation involves some specific rules that make calculations easier. One of these is the product of a power rule, which simplifies the work significantly. But, understanding the core rule is key.For any base \(a\) raised to an exponent \(m\), the rule of exponents tells us:

• The expression \(a^m\) means multiplying base \(a\) by itself \(m\) times.
• For instance, in \((-3)^4\), the computation is \((-3) \times (-3) \times (-3) \times (-3)\).
• If the exponent is positive, continue multiplying. If it's zero, the answer is typically \(1\) (as long as the base is not zero).

Each multiplication can affect the result's sign, especially with negative bases. Thus understanding these rules also helps predict if the result will be positive or negative.

Even Power

The concept of even power plays a critical role in determining the outcome's sign when the base is negative. An important rule is:- Any negative base raised to an even power will result in a positive number.For example, \((-3)^4\) leads to multiplying \(-3\) by itself four times. To comprehend why the result remains positive:

• Multiply the base in pairs:
  • First pair: \((-3) \times (-3) = 9\)
  • Second pair again: \(9 \times 9 = 81\)
• The even power means every two negative multiplications result in a positive product.

Thus, knowing about even and odd powers helps predict whether the product of repeated negative multiplication results in a positive or negative answer.