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Chapter 1: Problem 31

Evaluate. $$-2^{5}$$

Short Answer

Expert verified
The expression \(-2^5\) means "negative two raised to the power of five". To evaluate this, we multiply the base (-2) by itself four more times: \((-2) \times (-2) \times (-2) \times (-2) \times (-2)\). This results in \(16 \times -2\), which equals \(-32\). Therefore, \(-2^5 = -32\).

Step by step solution

01

Understand the expression

This expression can be read as "negative two raised to the power of five". The base of the expression is -2, and the exponent is 5. The exponent indicates that we need to multiply the base by itself four more times to calculate the result.
02

Perform the multiplication

To evaluate the expression, we need to multiply -2 by itself four more times: \((-2) \times (-2) \times (-2) \times (-2) \times (-2)\)
03

Multiply the numbers

Now we can multiply the pairs of numbers: \((-2 \times -2) \times (-2 \times -2) \times -2\) \(4 \times 4 \times -2\) \(16 \times -2\)
04

Find the result

Finally, we multiply the last two numbers: \(16 \times -2 = -32\) The result of the expression \(-2^5\) is \(-32\).

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Key Concepts

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

Understanding Exponents
An exponent indicates how many times a number, known as the base, should be multiplied by itself. In our exercise, the exponent is 5, and the base is -2. Let's break it down:
  • The base **-2** is not simply multiplied by 5; rather, it is multiplied by itself **5 times**.
  • The **exponent** (5) tells us **how many times** to multiply the base by itself.
  • Thus, in mathematical terms, \( (-2)^5 = (-2)\times(-2)\times(-2)\times(-2)\times(-2) \).
Handling expressions with exponents can seem complex at first, but remembering that it鈥檚 just repeated multiplication can simplify the process.
Working with Negative Numbers
Negative numbers are integers less than zero. They are opposite to positive numbers, and thus, have distinct properties:
  • Multiplying two **negative numbers** results in a **positive number**.
  • Multiplying a **positive number** with a **negative number** results in a **negative number**.
When you evaluate \((-2)^5\), you are multiplying **negative numbers** together multiple times:
  • First, multiply: \((-2)\times(-2) = 4\)
  • Again, since we have another **-2**, follow it with: \(4\times(-2) = -8\)
It鈥檚 important to remember the sign changes with each multiplication.
Understanding Multiplication
Multiplication is a basic mathematical operation that simplifies the addition of the same number multiple times.
  • It's denoted by a \( \times \) or \( * \) symbol.
  • For negative numbers, each multiplication step requires careful attention to signs, as seen with: \((-2)\times(-2)\)
When repeated with the same set of numbers, as in **exponential expressions**, multiplication becomes more complex. It requires sequential steps to simplify and calculate.Hence, in \((-2)^5\), our repeated multiplications highlighted the importance of maintaining correct results at each step.
Order of Operations
The order of operations is crucial in mathematics to ensure expressions are evaluated correctly. It is often remembered by the acronym PEMDAS:
  • **P**arentheses
  • **E**xponents
  • **M**ultiplication and **D**ivision (from left to right)
  • **A**ddition and **S**ubtraction (from left to right)
In our task, evaluating \(-2^5\), we focus first on the exponent. The minus sign does not affect the order, because the exponent operation is prioritized.After solving the exponent, \((2^5 = 32)\), the result is then affected by the negative sign to give the final result: \(-32\). This example clearly shows how ignoring the order can lead to different and incorrect results. Reviewing PEMDAS regularly helps prevent such errors.

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Most popular questions from this chapter

Write a mathematical expression for each and simplify. Two-thirds of \(-27\)

Write a mathematical expression for each and simplify. Twice the sum of 18 and \(-31\)

Use the order of operations to simplify. $$10-2(1-4)^{3} \div 9$$

Use the order of operations to simplify. $$7+8(-5)$$

Objectives 3 and 4 A welder must construct a beam with a total length of \(32 \frac{7}{8}\) in. If he has already joined a \(14 \frac{1}{6}\) -in. beam with a \(10 \frac{3}{4}-i n .\) beam, find the length of a third beam needed to reach the total length.
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