Problem 121 - A Graphical Approach to College Algebra

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$V=e^{3} \quad \text { for } e$$

Step-by-Step Solution

Answer

The solution is \( e = \sqrt[3]{V} \).

Step 1: Understand the equation

The given equation is \( V = e^3 \), and we need to solve for \( e \). This means we want to express \( e \) in terms of \( V \).

Step 2: Isolate the variable 'e'

The current equation is \( V = e^3 \). To isolate \( e \), we take the cube root of both sides of the equation. This gives us \( e = \sqrt[3]{V} \).

Step 3: Consider the implications of the cube root

When taking the cube root in real numbers, the solution can be negative or positive. However, for cube roots, we do not typically use \( \pm \) because the cube root of a negative number is real and negative. Hence, \( e = \sqrt[3]{V} \) is the complete expression.

Key Concepts

Exploring Cube Roots

Cube roots are mathematical operations that find the value which, when multiplied by itself three times, results in the original number. For example, if you have a number like 8, the cube root would be 2 because

• 2 × 2 × 2 = 8

Cube roots are denoted by the symbol \( \sqrt[3]{} \). Understanding cube roots is essential, especially since they appear in various mathematical problems, such as equations involving volume, as seen in the equation \( V = e^3 \).
Unlike square roots, cube roots of even negative numbers yield real and negative results. This property of cube roots is convenient as it doesn't require the use of the plus-minus (\( \pm \)) symbol, simplifying calculations.

The Art of Variable Isolation

Variable isolation is a fundamental skill in algebra that involves manipulating an equation to get one variable alone on one side of the equation. In an equation like \( V = e^3 \), isolating the variable \( e \) requires reversing the operations to solve for \( e \).
First, identify the operation being applied to the variable. In our case, \( e \) is being cubed. To counter this, apply the inverse operation: taking the cube root. By doing so, the original equation modifies to \( e = \sqrt[3]{V} \). Keep in mind:

• To isolate a variable effectively, always perform the same mathematical operation on both sides of the equation.
• The goal is to "undo" the operations applied to the variable.
• Isolate variables step-by-step when more complex operations are involved.

With practice, you'll become more efficient at spotting and applying these techniques to solve equations.

Understanding Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division). In our exercise, the expression \( V = e^3 \) involves:

• Variables such as \( V \) and \( e \).
• The operation of exponentiation, represented by \( e^3 \).

Algebraic expressions do not always constitute complete equations, but they form the backbone of many equations encountered in mathematics. Grasping the structure of algebraic expressions helps you understand how different elements interact.
Each component of an algebraic expression plays a vital role, and manipulating these components correctly can help solve equations. When simplifying or solving expressions, remember to:

• Follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Pay close attention to coefficients and variable powers.

A strong foundation in understanding algebraic expressions will set you up for success in more advanced mathematical concepts.