Problem 54 - Algebra A Combined Function

Question

Perform the indicated operation. \((-2)^{5}\)

Step-by-Step Solution

Answer

The result of \((-2)^5\) is \(-32\).

Step 1: Understand the Expression

The expression given is \((-2)^5\). This indicates that the base, \(-2\), is to be raised to the power of \(5\). This means we will multiply \(-2\) by itself a total of \(5\) times.

Step 2: Identify the Base and Exponent

In the expression \((-2)^5\), \(-2\) is the base, and \(5\) is the exponent. This tells us that the operation involves multiplying \(-2\) by itself 5 times.

Step 3: Multiply Step by Step

Compute the power step-by-step. Start by multiplying: 1. \(-2 \times -2 = 4\) 2. \(4 \times -2 = -8\) 3. \(-8 \times -2 = 16\) 4. \(16 \times -2 = -32\)The final result is \(-32\).

Step 4: Verify the Sign

Given that the exponent \(5\) is odd, raising a negative number to an odd exponent will result in a negative product, confirming that our result of \(-32\) is correctly signed.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a '-' sign, like -1, -2, etc. Negative numbers can be a bit tricky because their rules differ from positive numbers.
When multiplying or working with negative numbers in exponents, remember:

• Multiplying a negative number by a negative number results in a positive product. For example, o is \(( -2 \times -2 = 4 \)
• Multiplying a negative number with a positive number results in a negative product. For instance, o is \(( -2 \times 3 = -6 \)
• Exponents can also affect the sign, which we'll explore further in this article.

Understanding how negative signs interact during multiplication helps in grasping problems involving powers and exponents.

Multiplication

Multiplication is an arithmetic operation used to calculate the total number of items in equal-sized groups. When multiplying whole numbers, you're essentially adding a number to itself repeatedly.
When dealing with the multiplication of integers:

• Two positive numbers multiplied together give a positive product.
• Two negative numbers multiplied together also give a positive product, since the negatives cancel out.
• A positive and a negative number multiplied give a negative product.

This is crucial to remember, especially when multiplying a series of numbers as seen in evaluating powers or exponents. The order and the sign of each number influence the outcome.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself. The exponent indicates how many times the base number is multiplied by itself. In the expression o is \( (-2)^{5} \), o is \( 5 \) tells you that o is \( -2 \)%20is%20multiplied%20

• The small number, written above and to the right of the base, is the exponent.
• Multiplying the base by itself according to the value of the exponent is crucial to solving exponential expressions.
• Negative bases with odd exponents result in negative numbers, while negative bases with even exponents result in positive numbers.

When working through the operations, ensure to carefully multiply while keeping track of the sign.

Powers

The concept of powers refers mainly to expressions like o is \( a^b \), where o is \( a \) is referred to as the base and o is \( b \) %20is%20the%20exponent.
Calculating powers easily follows these key insights:

• A power represents how many times a number (the base) is used in multiplication.
• The multiplication of equal numbers is compactly expressed using powers.
• Powers simplify calculations by providing a shortcut to lengthy multiplications.

In applications like the given exercise, the power \((-2)^5 = -32\), allows you to express the multiplication of five -2's in one expression. Using powers, the operation o is \( -2 \times -2 \times -2 \times -2 \times -2 \) %20is%20shortened%20to%20o is \( (-2)^5 \), %20saving%20time%20and%20avoiding%20mistakes.