Problem 83 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$3 x^{2} \text { when } x=7$$

Step-by-Step Solution

Answer

The evaluated expression for \(x=7\) in \(3x^2\) gives the result 147.

Step 1: Substitute the Given Value

Replace \(x\) in the expression \(3x^2\) with the given value. This results in the new expression \(3 \times 7^2\).

Step 2: Simplify Exponent Expression

Simplify \(7^2\) to get 49. This changes the expression into \(3 \times 49\).

Step 3: Perform Multiplication

Now, multiply 3 and 49 together. The result is 147.

Key Concepts

Substitution Method

The substitution method is an essential algebraic technique used to replace variables with numerical values. In this exercise, we're asked to evaluate the expression \(3x^2\) for \(x = 7\). To begin, identify the variable in the expression, which is \(x\) in this case. You replace every instance of \(x\) with the given number, 7.
This process helps transform an algebraic expression into a numerical one. So, updating our expression \(3x^2\) becomes \(3 \times 7^2\).
It's crucial to ensure you're substituting correctly to avoid mistakes early on. By using substitution, you make the expression easier to manage and move towards solving the problem with basic arithmetic operations.

Exponents

Exponents, also known as powers, are a way of indicating that a number should be multiplied by itself a certain number of times. In the expression \(3 \times 7^2\), \(7^2\) uses an exponent, showing that 7 should be multiplied by itself once:

• 7 raised to the power of 2 equals \(7 \times 7\).
• This simplifies to 49.

Understanding how to work with exponents is vital because it allows you to simplify complex expressions efficiently. Always perform exponent calculations before multiplications or additions when following order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)). This ensures accuracy in solving expressions.

Algebraic Simplification

Algebraic simplification involves making expressions easier to understand or solve. After we substitute \(x\) and handle the exponent in \(3 \times 7^2\), we have the simplified form \(3 \times 49\).
Simplification helps by reducing the number of operations you need to perform. In our example, the next simplification is straightforward multiplication:

• Multiply 3 by 49 to get 147.

Through simplification, we take initially complex expressions down to simpler arithmetic operations. This step is crucial to reach the final answer efficiently. It also aids in visualizing your steps clearly and helps to prevent errors.