Problem 4 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$x^{4}-3 \text { when } x=2$$

Step-by-Step Solution

Answer

The evaluated result of the given expression is 13.

Step 1: Identify the variable value

Identify the given value for the variable in the expression. Here, \( x = 2 \) is given.

Step 2: Substitute the value in the equation

Now, substitute the value of x into the given equation. This gives us \( (2)^{4} - 3 \).

Step 3: Simplify the Equation

Solve the expression by first calculating the exponentiation. This gives us \( 16 - 3 \).

Step 4: Final Calculation

Now, subtract 3 from 16 which will give the final answer as 13.

Key Concepts

Variable Substitution

Variable substitution is the process of replacing a variable in an algebraic expression with a given value.
It is a fundamental technique in algebra and is crucial for evaluating expressions.
In this exercise, we know that the value for the variable \( x \) is 2.
By substituting, we transform the expression \( x^4 - 3 \) into \( (2)^4 - 3 \).
This step guarantees that the expression now only contains numbers, making it ready for computation.

• First, identify the variable and its given value.
• Then, replace the variable in all the relevant places within the expression.
• This allows you to proceed with no unknowns left in the expression.

This simple yet crucial step forms the basis for further calculations in any algebraic problem.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent.
It tells us how many times to multiply the base by itself.
In our expression, the operation \( 2^4 \) is performed as part of simplifying the expression.
Here, 2 is the base, and 4 is the exponent.

• The base (2) is multiplied by itself for a total of four times: \( 2 \times 2 \times 2 \times 2 \).
• This results in the number 16.

Understanding exponentiation is critical, as it frequently appears in both basic and advanced mathematics.
It is helpful to remember that exponentiation follows the order of operations, which dictates that it should be performed before addition or subtraction.

Simplification

Simplification is the process of breaking down an algebraic expression into its simplest form.
After substituting the variable, our expression became \( 16 - 3 \).
The simplification step completes the arithmetic operations to reach a single numerical result.

• First, perform any exponentiations in the expression.
• Then carry out any remaining additions or subtractions.

For our evaluated expression, once we compute \( 16 - 3 \), the simplest form is 13.
Simplification is an essential skill that allows us to clearly see the result of our calculations.
It involves applying the order of operations and reduces possible errors by organizing the steps appropriately.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and mathematical operators (such as addition, subtraction, and exponentiation).
It represents a particular value or set of values.
In our exercise, the given algebraic expression is \( x^4 - 3 \).

• Variables are used to stand in for unknown or variable quantities.
• Coefficients are numbers that multiply the variables.
• Constants are fixed numbers within the expression.

This specific expression is a polynomial with a term of \( x^4 \), indicating a fourth-degree polynomial.
Algebraic expressions can be manipulated through various mathematical operations, including the substitution and simplification we've discussed.
Mastery of algebraic expressions is essential as they form the groundwork for more complex mathematics and applications in science and engineering.