Problem 50 - Intermediate Algebra

Question

Evaluate each expression. See Example \(6 .\) $$ (-4.6)^{2} $$

Step-by-Step Solution

Answer

The result of \((-4.6)^2\) is 21.16.

Step 1: Understand the Expression

The expression given is \((-4.6)^2\), which means we are squaring the number \(-4.6\). This involves multiplying \(-4.6\) by itself.

Step 2: Perform the Multiplication

To square \(-4.6\), calculate \(-4.6 \times -4.6\). When you multiply two negative numbers, the result is positive. Therefore, calculate the product: \[ (-4.6) \times (-4.6) = 21.16 \]

Step 3: Conclude with the Result

The result of squaring \(-4.6\) is \(21.16\). You have evaluated the expression \((-4.6)^2\).

Key Concepts

Squaring Negative Numbers

Squaring a negative number might seem confusing at first, but it's a straightforward process. The term "squaring" refers to multiplying a number by itself. When the number is negative, like \(-4.6\), the rule of multiplication you need to remember is: multiplying two negative numbers results in a positive number.
When you square a negative number, say \((-4.6)^2\), the expression represents \(-4.6 \times -4.6\).

• The first negative sign from \(-4.6\) cancels out with the second negative sign from the other \(-4.6\), making the result positive.
• Therefore, \((-4.6)^2 = 21.16\). This exemplifies that squaring any negative number will yield a positive outcome.

The squaring of negative numbers is a classic example of this multiplicative property and serves as a reminder of how signs interact during multiplication.

Integer Multiplication

Getting familiar with integer multiplication is essential, especially when dealing with expressions that include negative numbers. The rules for multiplying integers include understanding how to handle the signs.

• Two positive numbers multiplied together yield a positive number.
• A positive number multiplied by a negative number gives a negative result.
• Two negative numbers multiplied will always result in a positive number, like in our example of \(-4.6 \times -4.6\) resulting in \(21.16\).

Mastering integer multiplication also aids in simplifying complex mathematical expressions and ensures accurate problem-solving across math disciplines. Knowing these foundational rules primarily helps to avoid mistakes made due to the signs of integers.

Evaluating Expressions

An expression in mathematics is a combination of numbers and operators that you work through to find a result. Evaluating expressions involves organizing these components logically to compute a solution. Let's break down the evaluation process:

• First, identify all the elements involved, including operations such as addition, subtraction, multiplication, or division.
• Apply the respective mathematical operations correctly. In our case, the task was to square the negative number \(-4.6\) by multiplying it by itself.
• Finally, perform the arithmetic calculations to reach a result. For \((-4.6)^2\), we multiplied and concluded with a positive \(21.16\).

Evaluating expressions accurately is crucial for problem-solving in math and lays the groundwork for understanding more complex equations and real-world scenarios. The key is always to follow the sequence of operations correctly to arrive at the right answer.