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Chapter 1: Problem 57

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

Short Answer

Expert verified
The evaluated expression is 343

Step by step solution

01

Substitute the given value into the expression

Replace \(b\) with 7 in the expression \(7b^{2}\) leading to the new expression \(7 * (7)^{2}\)
02

Perform exponentiation

Calculate \(7^2 = 49\), leading to the new expression \(7 * 49\)
03

Perform multiplication

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檝e 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鈥檚 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 \(343\).

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Most popular questions from this chapter

Match the problem with the formula needed to solve the problem. Then use the Guess, Check, and Revise strategy or another problem-solving strategy to solve the problem. Area of a rectangle \(\quad A=l w \quad\) Distance \(\quad d=r t\) Simple interest \(\quad I=P r t \quad\) Volume of a cube Temperature \(\quad C=\frac{5}{9}(F-32)\) Surface area of a cube \(S=6 s^{2}\) How long must \(\$ 1000\) be invested at an annual interest rate of \(3 \%\) to earn \(\$ 300\) in simple interest?

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