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Chapter 2: Problem 50

Compute the exact value. . \((-3)^{4}\)

Short Answer

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

Step by step solution

01

Understand the Problem

You are asked to find the exact value of \((-3)^{4}\). This means we will multiply the number -3 by itself four times.
02

Multiply the Base Number

Multiply the base number, -3, by itself. Begin with the first multiplication:\(-3 \times -3 = 9\).
03

Continue Multiplication

Now, take the result from step 2 and multiply it again by -3:\(9 \times -3 = -27\).
04

Complete the Exponentiation

Finally, multiply -27 by -3, which is the fourth multiplication:\(-27 \times -3 = 81\).
05

Conclusion

The calculation is complete and the final result of \((-3)^{4}\) is 81.

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Key Concepts

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

Multiplication
Multiplication is a fundamental mathematical operation where we find the total of one number added up a certain number of times. In this context, the task was to compute (-3)^4, which meant multiplying -3 by itself four times. Think of multiplication as repeated addition; for example, 3 multiplied by 4 means adding 3 together four times (3 + 3 + 3 + 3), resulting in 12.
  • The number being multiplied is known as the "multiplier." In our example, that's -3.
  • The number of times it is multiplied is called the "multiplicand," in this case, 4.
  • When multiplying multiple negative numbers, remember that an even count of negative numbers results in a positive product because the negatives "cancel out." For (-3)^{4}, as the negative sign is multiplied an even number of times, the result is positive.
Negative Numbers
Negative numbers are values less than zero that express the idea of a deficiency or loss. They can change the nature of a multiplication result significantly.
When multiplying with negative numbers, the sign of the final product depends on the number of negative factors:
  • If you have an even number of negative numbers in your multiplication, the result is positive. For instance, (-3)^{4} equals a positive number because four negative signs (an even count) cancel each other out to make a positive.
  • Conversely, if there is an odd number of negative factors, the final product comes out negative.
These principles are crucial in understanding how negativity affects multiplication outcomes.
Order of Operations
Order of operations is a rule that defines the correct sequence to evaluate a mathematical expression. This ensures everyone can understand and arrive at the same result for mathematical calculations.
The order can be remembered with the acronym PEMDAS:
  • P for Parentheses: Simplify expressions inside parentheses first (or brackets).
  • E for Exponents: Resolve exponents (such as squaring a number) next.
  • M/D for Multiplication/Division: Handle these operations from left to right as they appear in the expression.
  • A/S for Addition/Subtraction: Perform these last, also from left to right.
In our example of computing (-3)^{4}, we focus on the "E" in PEMDAS鈥攅xponents鈥攂y concentrating first on repeatedly multiplying -3 until the task is complete. Following these rules aids in organizing calculations and achieving the correct solution.

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