Problem 65 - Algebra A Combined Function

Question

Perform the indicated operation. \((-3)^{4}\)

Step-by-Step Solution

Answer

The answer is 81.

Step 1: Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. The expression \((-3)^{4}\) means -3 is being multiplied by itself 4 times.

Step 2: Multiply Base by Itself

Now, let's break down the operation:First, multiply -3 by itself to start with the first two factors:\((-3) \times (-3) = 9\).

Step 3: Continue Multiplying

Next, continue with the resulting product:Multiply 9 by -3:\(9 \times (-3) = -27\).

Step 4: Final Multiplication

Finally, multiply the last factor:\(-27 \times (-3) = 81\).This completes the operation because we've multiplied -3 by itself 4 times in total.

Step 5: Final Answer

The value of \((-3)^{4}\) is 81.

Key Concepts

Powers of Negative Numbers

When dealing with powers of negative numbers, you might wonder if the result will be positive or negative. This largely depends on the exponent. If the exponent is an even number, like 2, 4, or 6, the result of a negative base raised to that power is positive. If the exponent is odd, such as 1, 3, or 5, the result remains negative. This rule is rooted in the way multiplication works:

• Negative times negative gives a positive.
• Negative times positive remains negative.

So, in the expression \((-3)^{4}\), since the exponent 4 is even, multiplying -3 by itself four times ends up as a positive number, specifically 81.

Repeated Multiplication

Understanding repeated multiplication is key when working with exponents. An exponent essentially tells you how many times to multiply the base by itself. For example, \((-3)^{4}\) indicates multiplying -3 by itself four times. This is broken down into smaller steps such as:

• \((-3) imes (-3) = 9\),
• \(9 imes (-3) = -27\),
• \(-27 imes (-3) = 81\).

Each multiplication step shows how the negative base affects the outcome. By following these steps carefully, you can see how the repeated multiplication of a negative number eventually leads to a final result.

Mathematical Operations

Mathematical operations involve rules and techniques that help simplify and solve expressions. When performing operations with exponents, remember the following:

• Order of operations (PEMDAS/BODMAS) suggests that exponentiation appears high in precedence relative to other operations like multiplication and addition.
• Multiplying numbers, particularly negatives, requires attention to signs, as they determine if the result will be positive or negative.

In the expression \((-3)^{4}\), you apply multiplication repeatedly following the rule that a negative multiplied by a negative is positive, ensuring that the final result is 81.

Algebra Problem Solving

Successfully solving algebra problems often involves breaking them down into manageable steps. The goal is to simplify the expression thoroughly.In this exercise, you follow a sequence of steps to solve \((-3)^{4}\). This process includes:

• Identifying the base and exponent, recognizing the need to multiply the number by itself a specified number of times.
• Calculating each multiplication step by step instead of trying to do it all at once.

By tackling each component of the problem individually, you ensure accuracy and make solving the problem more approachable. Understanding each part helps build confidence in not just solving similar problems but also in comprehending the underlying concepts of algebra.