Problem 13 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

16

Step 1: Understand the Expression

The expression \((-4)^2\) tells us to take the number -4 and raise it to the power of 2.

Step 2: Break Down the Exponentiation

Raising a number to the power of 2 means multiplying the number by itself. Therefore, \((-4)^2\) is the same as \((-4) \times (-4)\).

Step 3: Perform the Multiplication

Multiply -4 by -4. The multiplication of two negative numbers results in a positive number. Therefore, \((-4) \times (-4) = 16\).

Key Concepts

negative numbers

Negative numbers can sometimes be confusing. A negative number is any number less than zero. You'll often see it with a minus sign in front of it. For example, -4 is a negative number because it is 4 units less than zero.
Understanding how negative numbers work with other operations is crucial.

• Adding a negative number is like subtracting its absolute value. For instance, 5 + (-3) is the same as 5 - 3.
• Subtracting a negative number makes the number positive. For example, 5 - (-3) equals 5 + 3.
• Multiplying or dividing two negative numbers gives a positive result. Multiplying or dividing a negative number by a positive number gives a negative result.

In the exercise \((-4)^2\), we are dealing with the multiplication of a negative number by itself.

multiplication

Multiplication is one of the basic operations in mathematics. When you multiply, you are essentially adding a number to itself a certain number of times. Here are some important points:

• Multiplication of two positive numbers results in a positive number. For example, 2 \times\ 3 = 6.
• Multiplication of a positive number and a negative number results in a negative number. For example, 2 \times\ (-3) = -6.
• Multiplication of two negative numbers results in a positive number. For instance, (-4)\times\ (-4) = 16.

In the exercise, we used multiplication to evaluate \((-4)^2\), which translates to \((-4) \times\ (-4)\). Knowing how to multiply negative numbers correctly helps in simplifying expressions.

simplifying expressions

Simplifying expressions is about making a math problem easier to understand and solve. It often involves combining like terms or breaking down complex expressions into simpler components. In the exercise \((-4)^2\), we simplify the expression as follows:
First, we recognize that \((-4)^2\) means \(-4 \times\ -4\). Using our knowledge of negative numbers and multiplication, we can work through the steps:

• Rewrite \((-4)^2\) as \(-4 \times\ -4\).
• Use the rule that multiplying two negative numbers results in a positive number.
• Calculate the product: \(-4 \times\ -4 = 16\).

By following these steps, you turn a potentially tricky problem into a straightforward one. Always remember to break down expressions and use the fundamental rules of arithmetic to guide you.