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Chapter 2: Problem 33

Evaluate. $$(-2)^{4}$$

Short Answer

Expert verified
So, to evaluate \((-2)^4\), we simply need to multiply the base, -2, times itself four times: \((-2) \times (-2) \times (-2) \times (-2)\). This results in the final answer of \(\boxed{16}\).

Step by step solution

01

Understand the meaning of the exponent

In this expression, \((-2)^4\), the exponent is 4. An exponent is a shorthand way to represent repeated multiplication. So, we need to multiply the base, -2, times itself four times.
02

Set up the expression

We can rewrite the expression as follows: \[ (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) \]
03

Multiply the numbers

Now we will simply multiply the numbers: \[ (-2) \times (-2) = 4 \] And then, we multiply the result by the next -2: \[ 4 \times (-2) = -8 \] Lastly, we multiply the result by the final -2: \[ -8 \times (-2) = 16 \]
04

Write the final answer

So, \((-2)^4\) equals 16. The answer is 16.

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation used to represent repeated multiplication of a number by itself. It involves two main components: the base and the exponent. In the expression \((-2)^4\), \(-2\) is the base and 4 is the exponent. This tells us that we need to multiply \(-2\) by itself four times.

When handling exponents, it's important to remember:
  • An exponent of 1 means the number is taken as it is. So, \((-2)^1 = -2\).
  • An exponent of 0 results in the value of 1, regardless of the base (except for the base being zero).
  • Higher exponents mean more instances of the base being multiplied.
Therefore, understanding exponentiation is key to simplifying and solving such expressions effectively.
Integer Exponents
Integer exponents are essential in mathematics, allowing us to perform repeated multiplication efficiently. Integer refers to whole numbers, which can be positive, negative, or zero. In relation to exponents:

  • Positive integer exponents denote how many times the base is multiplied by itself. For example, \(3^2\) equals 3 multiplied by itself, or 9.
  • Negative integer exponents are used to represent division or the reciprocal of the base raised to the corresponding positive exponent. For instance, \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).
  • Zero as an exponent results in a value of 1. So, \(5^0 = 1\).
In \((-2)^4\), the exponent is a positive integer 4, indicating that \(-2\) should be multiplied by itself four times. This illustrates the straightforward application of positive integer exponents.
Negative Numbers Multiplication
Multiplying negative numbers follows specific and consistent rules that help determine whether the product will be positive or negative:

  • When multiplying two negative numbers, the result is positive. For instance, \(-2 \times -2 = 4\).
  • When a negative number is multiplied by a positive number, the result is negative. Example: \(-2 \times 3 = -6\).
  • If the number of negative factors in a multiplication is even, the product is positive. Conversely, if it's odd, the product is negative.
In our given expression \((-2)^4\), since we are multiplying four negative numbers, which is an even count, the result is a positive number. Thus, step-by-step, \(-2 \times -2 = 4\), followed by \(4 \times -2 = -8\), and finally \(-8 \times -2 = 16\). This demonstrates how the rule about negative number multiplication is applied to calculate the correct output.

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Most popular questions from this chapter

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{2 \times 10^{1}}{5 \times 10^{4}}\)

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