Problem 33 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for then given value of the variable. \(x^{3}\) when \(x=5\)

Step-by-Step Solution

Answer

The value of the expression \(x^{3}\) when \(x=5\) is 125.

Step 1: Understand the Given Expression and Value

The given expression to be evaluated is \(x^{3}\) and the given value of the variable \(x\) is 5.

Step 2: Substitute the Given Value

Substitute the given value, \(x=5\), into the expression \(x^{3}\). So, the new expression becomes \(5^{3}\).

Step 3: Calculate the Expression

Calculate the value for the expression \(5^{3}\). The value of \(5^{3}\) based on the rules of exponents is \(5*5*5\) which is 125.

Key Concepts

Exponents

Exponents play a crucial role in algebra and beyond, denoting the power to which a number is raised. They simplify the representation of multiplication of the same factor multiple times. For example, an expression like \(5^3\) indicates that the number 5 is multiplied by itself three times, which translates to \(5 \times 5 \times 5 = 125\).

Understanding exponents involves recognizing the base, in this case, 5, and the exponent or power, which is 3. Exponents also follow specific rules that make calculations more manageable. For instance, any nonzero number raised to the zero power is always 1, while a number raised to the power of 1 is the number itself. Grasping these rules is essential for evaluating more complex algebraic expressions efficiently.

Substitution Method

When evaluating algebraic expressions, the substitution method is a fundamental approach that involves replacing the variable in an expression with its given numerical value. In the example of \(x^3\) when \(x=5\), the substitution method is applied by taking the 'x' in the expression and substituting it with 5, resulting in the new expression \(5^3\).

Proper substitution is key to obtaining the correct evaluation of an expression. This method is not only useful in algebra but also forms the basis for solving equations in calculus and other areas of advanced mathematics. It's important to perform substitution carefully, especially when dealing with complex expressions involving more than one variable or different operations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. For instance, in the expression \(x^3\), 'x' is a variable that represents an unknown value, and the cubed symbol (the exponent) notifies us to multiply 'x' by itself three times.

An important aspect of understanding algebraic expressions is recognizing that they are not equations since they do not contain an equals sign; rather, they represent values that can be calculated when the variables are given specific numerical values. Developing the skill to manipulate these expressions is vital for solving algebraic problems, including those found in higher-level math and scientific fields.