Problem 25 - Algebra 1: Concepts and Skills

Question

Find the product. \(-(7)^{2}\)

Step-by-Step Solution

Answer

\(-\(7^{2}\) = -49

Step 1: Calculate the square of 7

To calculate the square of 7, multiply 7 by itself. So, \(7^{2} = 7 \times 7 = 49\).

Step 2: Apply the negative sign

Now, apply the negative sign to the result obtained, so -\(7^{2} = -49\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-). In mathematics, negative numbers are as important as positive numbers and help in various calculations. They can often be thought of as the opposite of positive numbers. For example:

• While +3 signifies a profit, -3 indicates a loss.
• In temperature, +10°C is a rise, whereas -10°C is a drop.

When dealing with calculations involving negative numbers, it’s crucial to remember a few key rules:

• When you multiply or divide two negative numbers, the result is positive.
• If you multiply or divide a positive number with a negative number, the result is negative.
• Adding two negative numbers gives you a negative number, while subtracting a negative number is like adding its positive counterpart.

Understanding how negative numbers work is essential when solving mathematical problems like the original exercise, where we apply a negative sign after calculating the square of a positive number.

Exponents

Exponents represent repeated multiplication of a number by itself. They are written as a small number to the right and above the base number. For example, in the expression \(7^2\), 7 is the base, and 2 is the exponent. This expression means that 7 is multiplied by itself: \(7 \times 7 = 49\).
The exponent indicates how many times the base is used as a factor. It simplifies expressions and makes calculations more concise. Here are some basic rules of exponents:

• \(a^1 = a\). Any number to the power of 1 is itself.
• \(a^0 = 1\). Any number to the power of 0 is 1 (except for 0^0 which is undefined).
• To multiply powers with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).

Understanding exponents is crucial in operations involving powers, like in this exercise, where squaring the number was the first step in solving the problem.

Multiplication

Multiplication is a mathematical operation that combines groups of equal size. It is one of the four basic operations in arithmetic, along with addition, subtraction, and division.
In multiplication, you combine numbers to find the total number of objects in several groups. For instance, if you have 3 groups with 4 apples each, multiplication helps you find the total: \(3 \times 4 = 12\).
In the context of the given exercise, multiplication is used to find the square of a number. Squaring a number involves multiplying the number by itself. When you calculate \(7^2\), you're simply doing \(7 \times 7\), resulting in 49.

• Commutative Property: \(a \times b = b \times a\). The order of multiplication doesn't affect the result.
• Associative Property: \((a \times b) \times c = a \times (b \times c)\). You can regroup numbers, and the result stays the same.
• Identity Property: \(a \times 1 = a\). Any number multiplied by 1 is itself.

These properties make multiplication a flexible and versatile operation, especially useful when dealing with problems that involve exponents and negative numbers.