Problem 57 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ 7 b^{2} \text { when } b=7 $$

Step-by-Step Solution

Answer

The evaluated expression is 343

Step 1: Substitute the given value into the expression

Replace \(b\) with 7 in the expression \(7b^{2}\) leading to the new expression \(7 * (7)^{2}\)

Step 2: Perform exponentiation

Calculate \(7^2 = 49\), leading to the new expression \(7 * 49\)

Step 3: Perform multiplication

Multiply 7 by 49 to get the final result, 343.

Key Concepts

Substitution in Algebraic Expressions

Substitution is an essential concept in algebra that involves replacing variables in an expression with specific values. This is often one of the first steps taken when evaluating an expression for a given variable. In our exercise, the expression we started with was \(7b^2\), where we needed to evaluate it for \(b = 7\). To perform substitution correctly:

• Identify the variable in the expression, which in this case is \(b\).
• Replace the variable with the given number. Here, we replace \(b\) with \(7\), transforming \(7b^2\) into \(7 \times (7)^2\).

Using substitution helps simplify the problem and prepare it for further calculations, such as exponentiation and multiplication, which are the next steps in solving the expression.

Understanding Exponentiation

Exponentiation is the process of raising a number to a power, which means multiplying the number by itself a certain number of times. In our expression \(7 \times (7)^2\), the exponentiation part is \(7^2\). This means we multiply \(7\) by \(7\).

• The base is the number that gets multiplied, which is \(7\) here.
• The exponent tells us how many times the base is used as a factor. With \(7^2\), the number 2 is the exponent.

Calculating \(7^2\) gives us \(49\). It is crucial to understand that exponentiation is not the same as multiplication; rather, it is a repeated multiplication. Grasping this concept helps in proceeding with further operations, knowing you’ve evaluated part of the expression correctly.

Mastering Multiplication

Multiplication is a fundamental arithmetic operation where we combine equal groups to find a total. After handling substitution and calculating exponentiation in our original expression, we have \(7 \times 49\) left to compute. This multiplication step is straightforward:

• You take the first number, which is \(7\), and multiply it by \(49\).
• Ensure each number is aligned correctly when calculating manually, or use a calculator for efficiency.
• \(7 \times 49\) equals \(343\).

Multiplication allows us to scale products efficiently, and it’s important to perform it with precision to ensure proper results. By following these steps, the solution to the expression \(7b^2\) for \(b = 7\) is completed correctly, yielding \(343\).