Problem 66 - Algebra 1

Question

Evaluate the expression for the given value of the variable. (Review 1.2) $$9 b^{2} \text { when } b=8$$

Step-by-Step Solution

Answer

The numerical value after evaluating the expression \(9b^2\) when \(b=8\) is \(576\).

Step 1: Substituting the given value

Replace \(b\) in the expression \(9b^2\) with the given value, \(8\). This results in the expression becoming \(9*8^2\). Note that in this context, the caret (^) is used to denote an exponent.

Step 2 : Evaluating the exponent

The first operation deriving from standard order of operations (PEMDAS/BODMAS) would be to evaluate \(8^2\). Squaring 8 yields \(64\). The expression simplifies further into \(9*64\).

Step 3: Multiplying the remaining numbers

The final operation constituting of multiplying \(9\) and \(64\) is executed, thus giving \(576\) as the final numerical value of the original expression.

Key Concepts

Order of Operations

When evaluating mathematical expressions, it's essential to follow the order of operations. This ensures accuracy and consistency across calculations. The order of operations is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)) or BODMAS (Brackets, Orders or Exponents, Division and Multiplication (left to right), Addition and Subtraction (left to right)).

• Parentheses or Brackets: Solve any calculations inside parentheses or brackets before dealing with anything else.
• Exponents or Orders: Work on any exponents or powers next.
• Multiplication and Division: Address these operations from left to right.
• Addition and Subtraction: Finally, carry out these operations from left to right.

In the given exercise, after substituting the value in place of the variable, our expression was left with an exponent and a multiplication to evaluate. By correctly applying the order of operations, you first solve the exponentiation before multiplying, ensuring the correct result.

Substitution

Substitution is a technique used in algebra where variables in an expression are replaced with their corresponding numerical values. It simplifies expressions by allowing us to compute their exact values. This practice is especially useful in word problems or where specific values for variables are provided.In the exercise given, substitution happened in the first step. The variable \(b\) in \(9b^2\) was replaced by its given value, 8.

• This resulted in a new expression: \(9 \times 8^2\).
• The use of substitution helps transition the expression from abstract to numerical, making subsequent calculations possible.

Correct substitution is crucial, as any error here leads to incorrect solutions in further steps.

Exponents

Exponents are a mathematical notation indicating the number of times a number, called the base, is multiplied by itself. It's an essential part of algebra and helps in simplifying repeated multiplication.In the expression \(9b^2\), \(b^2\) indicates that \(b\) (which is 8 in our case) is multiplied by itself. Here's how it works:

• Calculate \(8^2\), which means \(8 \times 8\).
• This computation results in 64, as it's the product of 8 multiplied by itself.

Understanding exponents is pivotal in solving many algebraic equations and expressions, as it directly impacts the order of operations. Properly evaluating the exponents before other operations prevents miscalculations and errors in the final result.