Problem 74 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(b^{2}\) when \(b=8\)

Step-by-Step Solution

Answer

The value of \(b^{2}\) when \(b=8\) is 64.

Step 1: Understand the operation

The operation we need to perform is squaring. In mathematics, squaring a number means multiplying the number by itself. Since here we have \(b=8\), we need to multiply 8 by itself.

Step 2: Perform the operation

Now, let's perform the squaring operation. We have \(b^{2}\), where \(b=8\), so performing the operation gives us \(8^{2} = 8 \times 8 = 64\).

Key Concepts

Squaring Numbers

When you encounter the term 'squaring' a number, what you're doing is multiplying that number by itself. The term 'square' comes from the geometric shape, which has equal sides. So when you 'square' a number, you are essentially determining the area of a perfect square with sides equal to the number in question.

For example, if you have the number 8 and want to square it, you would calculate it as follows:
\[ 8^2 = 8 \times 8 = 64 \].
This result, 64, is called the 'square' of 8. Squaring numbers is a fundamental operation in algebra that can be applied to both positive and negative numbers as well as zero. When dealing with negative numbers, remember that a negative times a negative gives a positive result, so squaring a negative number will always yield a positive outcome.

Exponents

An exponent refers to the number that tells how many times the base—a term that appears below the exponent—should be multiplied by itself. The operation is known as 'exponentiation'. Consider the example where you have an expression like \( b^2 \). Here, 2 is the exponent, and 'b' is the base. It signifies that the base ‘b’ should be used as a factor twice.

Working with Exponents

It's not just about squaring; exponents can be any positive or negative whole number, and the operation extends to these as well. For instance, \( b^3 \) means multiplying 'b' three times as \( b \times b \times b\). Evaluating exponent expressions requires careful consideration of the exponent values because they significantly influence the outcome.
Exponents have their own set of properties and rules which make calculations easier such as the product of powers rule, power of a power rule, and negative exponent rule.

Algebraic Operations

Dealing with algebra involves a variety of operations including addition, subtraction, multiplication, division, and exponentiation among others. Each of these operations has specific rules for how to apply them, especially when variables are involved.

In the context of our exercise, after understanding what squaring a number means and what exponents are, we apply that knowledge to perform the algebraic operation. The process is moved forward by substituting the value of the variable (in our case, 'b' equals 8) into the algebraic expression followed by the execution of the operation dictated by the exponent.
Therefore, to evaluate \( b^2 \) for \( b = 8 \), we replace 'b' with 8 and square the number, leading us to the solution \( 8^2 = 64 \). Algebraic operations often rely on the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure expressions are solved correctly and consistently.