Problem 13 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when \(x=3.\) $$ x+3 x^{4} $$

Step-by-Step Solution

Answer

The evaluated expression when \(x = 3\) is 246.

Step 1: Identify the Given Values

In this problem, you are given that \(x = 3\) and the expression to evaluate is \(x + 3x^4\).

Step 2: Substitute the Given Value into the Expression

Replace \(x\) in the expression \(x + 3x^4\) with the given value of 3. The expression then becomes: \(3 + 3*3^4\).

Step 3: Evaluate the Expression

First perform the operation in the parentheses, \(3^4 = 81\). After simplification, the expression becomes \(3 + 3*81\), which further simplifies to \(3 + 243\). Finally, add together the numbers to get the final answer: \(3 + 243 = 246\).

Key Concepts

Substitution Method

The substitution method is a crucial technique in mathematics, especially when evaluating variable expressions. At its core, substitution involves replacing a variable with a given numeric value.
In our exercise, we were given the expression \(x + 3x^4\) and the value \(x = 3\). By substituting 3 for every occurrence of \(x\) in the expression, it transforms into \(3 + 3 \cdot 3^4\). This step is essential to simplify and evaluate the expression accurately.
Using substitution, you're reducing variables into actual numbers, thus simplifying calculations. Remember to replace every instance of the variable to maintain consistency and accuracy.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a certain power. This is a crucial operation when dealing with expressions like \(3x^4\).
In our example, \(3^4\) means that the number 3 is multiplied by itself 4 times. So you perform the calculation:

• \(3 \times 3 = 9\)
• \(9 \times 3 = 27\)
• \(27 \times 3 = 81\)

Thus, \(3^4\) simplifies to 81. Exponentiation simplifies repeated multiplication, making it a powerful tool in expression evaluation.
Always perform exponentiation before other operations like multiplication or addition in an arithmetic expression, following the order of operations.

Simplification

Simplification of expressions is the process of reducing them to their simplest or most easily workable form. In mathematical expressions, it involves performing the operations indicated.
To simplify our expression after substitution, we handle the operation inside the power first: \(3 + 3 \cdot 81\).

• Calculate \(3 \cdot 81 = 243\).

The expression then simplifies to \(3 + 243\).
Simplification makes it easier to interpret and solve expressions. Always simplify step-by-step, following the appropriate order of operations to avoid mistakes.

Evaluation of Expressions

Evaluating an expression means finding its numerical value. This is where everything comes together.
After substituting and simplifying properly, the final step is to evaluate the simplified expression: \(3 + 243\).

• Add them up to get the result: \(246\).

Evaluation requires precision and careful execution of all the preceding steps.
Always double-check your work by following each step methodically, ensuring that each part of the expression is accounted for and accurately calculated.