Problem 38 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ c^{6} \text { when } c=2 $$

Step-by-Step Solution

Answer

The value of the expression \(c^{6}\) when \(c=2\) is 64.

Step 1: Substituting the value of c into the expression

We need to replace the variable \(c\) in the expression \(c^{6}\) with the given value, which is 2. So the expression becomes \(2^{6}\).

Step 2: Performing the Exponentiation

Now we will raise 2 to the power of 6. This means multiplying 2 by itself 6 times. The result is 64.

Key Concepts

Substituting Variables

When working with algebraic expressions like \( c^{6} \), the process of substituting variables is fundamental. It involves replacing the variable, in this case \( c \), with its given value. Think of the variable as a placeholder or a blank space waiting to be filled.

To effectively substitute a variable, you must ensure accuracy in replacing it wherever it appears in the expression. In our exercise, the given value for \( c \) is 2. Thus, we swap \( c \) with 2 throughout the expression, resulting in \( 2^{6} \). Proper substitution serves as the initial step that leads you to correct calculations and understanding complex algebraic operations to follow.

Exponentiation

Exponentiation is the process of raising a number to a certain power. In our example, the number 2 is raised to the power of 6, denoted as \( 2^{6} \). This mathematical operation is essentially a shorthand for repeating multiplication of the base number, which is 2 in our case, as many times as indicated by the exponent, here 6 times.

Remember, an exponent tells you how many times to use the number in a multiplication. It's not six twos added together; it's 2 multiplied by itself 5 more times after the first: \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \). Exponentiation is a powerful tool and understanding it is crucial as it forms a part of many algebraic and geometric calculations.

Algebraic Expressions

Algebraic expressions, like \( c^{6} \), represent a core component of algebra. They consist of variables and constants combined using mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Think of algebraic expressions as a way to generalize arithmetic by allowing us to work with unknown values.

These expressions can become quite complex with multiple terms and power, but the principles for evaluating them remain consistent. Understanding how to manipulate these expressions by performing operations in the correct order, applying properties of exponentiation, and simplifying results ultimately leads to mastering algebra and solving various mathematical problems.