Problem 30 - Elementary Algebra

Question

\((-7)^{2}\)

Step-by-Step Solution

Answer

49

Step 1: Identify the Expression

The given expression is \((-7)^{2}\). This indicates that -7 is raised to the power of 2, or squared.

Step 2: Perform the Squaring Operation

Squaring a number means multiplying it by itself. Therefore, calculate \(-7 \times -7\).

Step 3: Calculate the Result

When multiplying two negative numbers, the result is positive. Hence, \(-7 \times -7 = 49\).

Step 4: Write the Final Answer

Thus, \((-7)^{2} = 49\).

Key Concepts

Squaring

Squaring a number means raising it to the power of 2. In other words, you multiply the number by itself. For example, in the exercise, we have \((-7)^{2}\). This means we need to compute \(-7 \times -7\). Squaring is a very common operation in mathematics and plays a vital role in areas such as algebra, geometry, and physics.
When you square a number, whether it's positive or negative, the result is always non-negative. This is because a negative number times itself results in a positive number. Understanding squaring helps in solving quadratic equations, working with areas, and more.

Multiplication

Multiplication is one of the fundamental arithmetic operations. It can be thought of as repeated addition. When you multiply two numbers, you are essentially adding one number to itself a specified number of times. In the exercise, we need to compute \(-7 \times -7\).

• Product of Negatives: When you multiply two negative numbers, the result is positive. This is because each negative sign cancels out.
• Illustration: Think of \(-7 \times -7\) as adding \(-7\) a total of seven times but in a negative fashion. So, you're effectively removing seven groups of \(-7\) numbers, resulting in \(+49\).

Understanding multiplication is crucial for working with algebraic expressions, solving equations, and handling many real-life scenarios such as scaling recipes or calculating areas.

Negative Numbers

Negative numbers are numbers less than zero. They are typically used to represent values such as debts, temperatures below freezing, or elevations below sea level. In the exercise, we started with \-7^{2}\.

• Multiplication of Negatives: When you multiply two negative numbers, the product is positive. This principle is essential to understand as it frequently appears in algebra.
• Example: For instance, when you have \-7\ and \-7\ in \(-7 \times -7\), think of it as removing a negative (which is an addition) so the product becomes positive.

Grasping the concept of negative numbers helps with many aspects of math and practical situations, like calculating depth, understanding temperature variations, and more.