Problem 68 - Algebra 1

Question

Evaluate the expression for the given value of the variable. (Review 1.2) $$(7 x)^{3} \text { when } x=2$$

Step-by-Step Solution

Answer

2744

Step 1: Understand the expression

Here is a typical mathematical expression: \( (7x)^3 \). This expression means that the product of 7 and x is cubed. The value of the variable x is given as 2.

Step 2: Substitute in the given value

We substitute x with 2 in the original expression: \( (7 * 2)^3 \). Therefore, our expression becomes \( 14^3 \).

Step 3: Apply the exponent

After substitution, the expression becomes a simple exponentiation. We do the cube of 14: \( 14 * 14 * 14 = 2744 \).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. This means we multiply the base by itself as many times as indicated by the exponent. For example, in the expression \((7x)^3\), the entire product \(7x\) is the base and 3 is the exponent. The expression is read as "7x raised to the power of 3," or "7x cubed." However, this is not just limited to single numbers; any expression that involves multiplication can be raised to a power.
To compute \((7 \times 2)^3\), you first calculate \(7 \times 2\), which equals 14. Then, you perform exponentiation on 14 by multiplying it by itself three times:
\[ 14 \times 14 \times 14 = 2744. \]

• Base: A number or expression to be multiplied by itself.
• Exponent: Indicates how many times the base gets multiplied by itself.
• Cubing: A special case of exponentiation where the exponent is 3.

Substitution

Substitution is a simple process of replacing variables with their numerical values in an algebraic expression. It serves as a crucial step towards solving expressions or equations because it allows us to work with specific numbers. In our example, the variable \(x\) is given the value 2. To substitute \(x\) means to replace every occurrence of \(x\) with 2 in the expression.
After substituting \(x\) with 2, the expression \((7x)^3\) becomes \((7 \times 2)^3\). Now, you don’t have an unknown variable to worry about, making the arithmetic much clearer and easier to manage.
Steps for substitution in the given context:

• Identify the variable and its given value.
• Replace variable in the expression with that value.
• Re-calculate the expression without any variables.

Variable Evaluation

Variable evaluation involves determining the numeric value of an expression based on given values of variables. In this context, it is the final step where you evaluate the expression \((7x)^3\) using the provided value for \(x\). After substituting \(x = 2\) into the expression, we prepare it for final evaluation: \((7 \times 2)^3\). This is vital because it translates an abstract expression into a concrete number.
Once substitution is done and the expression becomes \(14^3\), evaluate it by applying exponentiation, resulting in 2744.
Variable evaluation steps encapsulated:

• Substitute the variable's value in the expression.
• Simplify the expression to eliminate variables.
• Calculate the resulting numeric expression.

This process turns an algebraic expression into a numerical result, solving the expression fully.