Problem 7 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The expression \((-2)^4\) evaluates to 16.

Step 1: Understand the Expression

The expression \((-2)^4\) means \(-2\) should be multiplied by itself 4 times.

Step 2: Write the Multiplication

Re-write \((-2)^4\) as a multiplication: \((-2) \times (-2) \times (-2) \times (-2)\).

Step 3: Multiply in Pairs

Calculate the product of the first pair: \((-2) \times (-2) = 4\).

Step 4: Continue Multiplying

Now multiply the result, 4, by the next pair: \(4 \times (-2) = -8\).

Step 5: Final Multiplication

Finally, multiply the result, -8, by -2: \((-8) \times (-2) = 16\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are similar to positive numbers but have a minus sign in front of them, indicating their position on the number line below zero. Understanding negative numbers is crucial in mathematics because they are often involved in operations where you are subtracting or working with losses and debts.
When we multiply two negative numbers, such as

• \((-2) \times (-2)\), the result is a positive number.

This can be counter-intuitive, but the rule is straightforward:

• A negative times a negative gives a positive result.
• A negative times a positive gives a negative result.
• A positive times a positive results in a positive number.

Understanding this helps avoid mistakes in calculations. Remember, the sign of a number greatly influences multiplication results, especially in expressions with exponents, like \((-2)^4\).
Being comfortable with how negative signs affect calculations will help in evaluating expressions involving powers and exponents.

Multiplication

Multiplication is a fundamental arithmetic operation used to calculate the total number of items when you have groups of the same size. In simpler terms, it's like repeatedly adding a number. For example, multiplying

• \(3 \times 4\) is the same as adding three 4s (\(4 + 4 + 4 = 12\)).

When dealing with powers, such as \((-2)^4\), multiplication becomes repetitive because it represents

• \((-2)\) being multiplied by itself four times.

In our specific exercise, the computation involves

• \((-2) \times (-2) \times (-2) \times (-2)\).

By grouping the multiplication in pairs, we get

• \((-2) \times (-2) = 4\), so each pair of negative two multiplies to give positive four.

Continuing this process leads to the overall result. Knowing how to use multiplication effectively helps simplify the process of working through exponentiation problems.

Powers

Powers, also known as exponents, describe how many times a number is multiplied by itself. In the expression

• \((-2)^4\), the number \(-2\) is the base, and 4 is the exponent.

The purpose of using powers is to shorten repeated multiplication, making calculations easier and expressions more concise. Powers follow the multiplication rule, but they have added rules due to their nature.
For instance, even though

• \((-2)^4\) involves negative numbers, it results in a positive number.

Why? Because multiplying an even number of negative factors always gives a positive result. In contrast, an odd number of negative factors would result in a negative outcome. Recognizing this difference is essential.
Remember that powers are a shorthand way of expressing long multiplication series efficiently, which is particularly useful in algebra and higher-level mathematics. Understanding how powers operate on numbers, especially negative ones, will improve your ability to tackle various math problems.