Problem 68 - Elementary and Intermediate Algebra

Question

Solve each equation for the specified variable or expression. $$ r=\sqrt[3]{\frac{A}{P}}-1 \text { for } P $$

Step-by-Step Solution

Answer

\( P = \frac{A}{(r+1)^3} \)

Step 1: Isolate the Cube Root Expression

The equation given is \( r = \sqrt[3]{\frac{A}{P}} - 1 \). To isolate the cube root, we first add 1 to both sides of the equation: \( r + 1 = \sqrt[3]{\frac{A}{P}} \).

Step 2: Remove the Cube Root

We need to eliminate the cube root to solve for \( P \). To do that, we will cube both sides of the equation: \( (r + 1)^3 = \frac{A}{P} \).

Step 3: Solve for the Variable P

Multiply both sides by \( P \) to remove the fraction: \( P(r + 1)^3 = A \). Then, divide both sides by \( (r+1)^3 \) to solve for \( P \): \( P = \frac{A}{(r+1)^3} \).

Key Concepts

Solving Equations

When we talk about solving equations in algebra, we mean finding the value of a variable that makes the equation true. Solving equations often involves performing the same operation on both sides of the equation until the variable is isolated.

Here's how it works: imagine the equation is like a balanced scale. To keep it balanced, whatever you do to one side, you must do to the other. This process often includes certain steps:

• Identifying the variable to solve for.
• Undoing any operations that involve the variable. This might include adding, subtracting, multiplying, dividing, or dealing with exponents and roots.
• Re-checking your solution by substituting it back into the original equation to make sure it satisfies the equation.

By following these steps, you can systematically approach any algebra problem and find the value of the unknown variable.

Variable Isolation

Variable isolation is a critical step in solving equations. It refers to the process of manipulating the equation to get the unknown variable by itself on one side of the equation. This is typically the goal when solving equations because it allows us to find the solution directly.

To isolate a variable, you should:

• Move terms containing the variable to one side of the equation.
• Move constants (numbers without variables) to the opposite side.
• Perform inverse operations—if something is added to the variable, you subtract; if it is multiplied, you divide, and so on.

Isolation of the variable makes the solution clear and straightforward. The key is to keep the equation balanced during the entire process.

Cube Root

Understanding how to work with the cube root is essential for solving some algebra problems. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Here's a quick example: the cube root of 8 is 2, because 2 \( \times \) 2 \( \times \) 2 = 8. In equation form, the cube root is represented as \( \sqrt[3]{x} \).

• To remove a cube root, you cube both sides of the equation. This operation is the inverse of taking the cube root.
• Cubing a number helps understand its relation by expanding it into a three-dimensional space, compared to squaring, which expands into two dimensions.

By mastering how to effectively manipulate cube roots, you can solve more complex algebraic equations and deepen your mathematical understanding.