Problem 39 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ w^{3} \text { when } w=13 $$

Step-by-Step Solution

Answer

The value of the expression \(w^{3}\) when \(w = 13\) is 2197.

Step 1: Understand the expression

In the expression \(w^{3}\), the exponent \(3\) indicates that the base \(w\) is to be multiplied by itself twice. So, \(w^{3}\) can be expanded to \(w * w * w\).

Step 2: Substitute the given value

Replace the variable \(w\) with the given value which is 13. The expression now becomes \(13 * 13 * 13\).

Step 3: Calculation

Continue by performing the multiplication operation. Multiply 13 by itself twice, which yields 2197.

Key Concepts

Understanding Variables

In mathematics, a variable is a symbol or letter that stands in for a number we don't know yet or can change. Variables are widely used to describe mathematical situations or relationships between different numbers or quantities. For example, the expression \(w^3\) includes the variable \(w\). Here, \(w\) can represent any number. Variables are essential for creating expressions that work for many scenarios, allowing us to solve various problems by substituting different values. This makes variables a powerful tool in mathematics.

The Art of Substitution

Substitution is the process of replacing a variable with a given number or another expression. In our exercise, we replace the variable \(w\) with the number 13, as specified. This turns the expression from \(w^3\) into \(13^3\).

• Identify the variable in your expression, which here is \(w\).
• Replace or substitute \(w\) with its given value of 13.

Substitution is a straightforward but crucial step in evaluating any mathematical expression that includes variables.

Mastering Multiplication

Multiplication is one of the fundamental operations in mathematics. In the context of exponents, it involves repeated multiplication of the base number. In our exercise, once the variable \(w\) is substituted with the number 13, we then perform the multiplication: \(13 \times 13 \times 13\). This step requires you to:

• Identify how many times to multiply the number based on the exponent, which is 3 in \(w^3\).
• Calculate each step, such as \(13 \times 13 = 169\), and then \(169 \times 13\).

Ultimately, performing these multiplications will yield the final result.

Evaluation of Expressions

Evaluation of expressions means finding the numerical value of an expression after substituting the variables with their given values. In the given exercise, you've replaced \(w\) with the number 13, and you've calculated \(13^3\) as 2197.

• Start by substituting the variable with its value.
• Perform the operation indicated by the expression (here it's multiplication following the power or exponent).
• Present the final number, which in this case is 2197.

Successful evaluation is about following these steps systematically to reach the correct result, ensuring you fulfill the criteria of the expression accurately.