Question

Evaluate the expression. \(b^{2}\) when \(b=8\)

Short answer

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

Step-by-step solution

Identify the expression

The expression provided is \(b^{2}\).

Identifying the given value

The value provided for \(b\) is 8.

Substituting in value

Substitute 8 for \(b\) in the expression \(b^{2}\) to get \(8^{2}\).

Evaluate the expression

Perform the calculation \(8^{2} = 64\).

Key concepts

Algebraic Expressions

When we talk about algebraic expressions, we are referring to a combination of numbers, variables (like the letter b in our exercise), and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions are the very backbone of algebra and provide a way to express general mathematical relationships.

An important aspect of algebraic expressions is understanding how to manipulate them according to the rules of algebra. For instance, one can simplify expressions, factor them, or expand them. When it comes to evaluating algebraic expressions, as in the provided exercise, our goal is to find the numerical value of the expression for given values of the variables involved.

Let's consider our example, where the expression is a simple one, consisting of a variable b raised to the power of 2, denoted as b². To evaluate this for b=8, we simply need to replace the variable b with 8 and compute the resulting numerical value, which is a key skill in algebra.

Exponentiation

Exponentiation is a mathematical operation where a number is multiplied by itself a certain number of times. The number that is being multiplied is called the base, and the number of times it is multiplied by itself is called the exponent. In our example, 8², 8 is the base and 2 is the exponent.

The result of exponentiation is known as a power. So, when we calculate 8², it means we are computing the second power of 8, which involves multiplying 8 by itself once because the exponentiation process has one implied multiplication already (8 x 8 = 64). It is crucial for students to remember that the exponent tells us how many times the base is used as a factor, which does not include the first occurrence of the base.

Understanding exponentiation is fundamental when working with algebraic expressions, as it allows us to represent and manage growth patterns, area calculations, and much more in a compact form.

Substitution in Algebra

Substitution is a technique often used in algebra to solve equations or evaluate expressions. It involves replacing variables with numbers or with other expressions. This process is critical when you're given a specific value for a variable, just like in the exercise with b=8, and you are asked to find out what the entire expression equals with that value.

In our step-by-step solution, we performed substitution by replacing the variable b with its given value, 8. The substitution makes the abstract expression concrete, and we can then carry out the necessary arithmetic operations to evaluate it.

Why is substitution important?

Substitution helps in solving for unknowns and in simplifying complex algebraic expressions. It also paves the way to solve systems of equations where one or more equations are used to find multiple variables. Substitution ensures that algebra remains not just a theoretical endeavor but a practical tool for solving real-world problems.