Problem 62 - Precalculus: Functions and Graphs

Question

Solve for the specified variable. $$A=B \sqrt[3]{\frac{C}{D}}-E \text { for } D$$

Step-by-Step Solution

Answer

\( D = \frac{CB^3}{(A + E)^3} \)

Step 1: Isolate the Term Involving D

Start by isolating the term involving \( D \) on one side of the equation. We have the equation \( A = B \sqrt[3]{\frac{C}{D}} - E \). Move \( E \) to the left side to get \( A + E = B \sqrt[3]{\frac{C}{D}} \).

Step 2: Divide by B

To further isolate the cube root term, divide both sides of the equation by \( B \). This gives us \( \frac{A + E}{B} = \sqrt[3]{\frac{C}{D}} \).

Step 3: Eliminate the Cube Root

To eliminate the cube root, raise both sides of the equation to the power of 3. This results in \( \left(\frac{A + E}{B}\right)^3 = \frac{C}{D} \).

Step 4: Solve for D

Now, solve for \( D \) by taking the reciprocal of both sides. This gives \( D = \frac{C}{\left(\frac{A + E}{B}\right)^3} \).

Step 5: Simplify the Expression

Simplify the expression: \( D = \frac{CB^3}{(A + E)^3} \). This is the simplified formula for \( D \).

Key Concepts

Isolating variables

To solve for a specific variable, such as in our case with "D," one of the key steps is isolating it on one side of the equation. Let’s simplify this idea using a simple strategy.

• Identify the terms involving the variable you want to solve for. Here, we're focusing on the term with "D", which is nested within a cube root.
• Rearrange the equation to get this term by itself on one side. This often involves adding or subtracting other terms from both sides of the equation. For instance, in the exercise, we move "E" to the left side of the equation to start the isolation process.
• Isolating a variable might also require dealing with coefficients or other operations (like division or multiplication) to further simplify the equation.

When you get the term by itself, it's much easier to manipulate the equation to solve for the desired variable. This foundational step sets the stage for further transformations and solutions.

Cube root

The cube root is a mathematical operation that undoes cubing. In our exercise, we have a cube root operation represented by \(\sqrt[3]{x}\), where "x" is the expression involving "C" and "D." Understanding the cube root is crucial for manipulating our equation.

• To eliminate a cube root in an equation, raise both sides to the power of 3. This will "undo" the cube root.
• The cube root of any number \(y\), when cubed, will give back the number "y": \((\sqrt[3]{y})^3 = y\).
• By removing the cube root from one side, the equation becomes simpler and able to further operations.

Cube roots are similar to square roots but relate to multiplying a number by itself three times. Understanding this concept empowers you to effectively simplify equations with cube roots.

Rational expressions

Rational expressions are fractions where the numerator and the denominator consist of polynomials. In the provided exercise, we encounter the rational expression \(\frac{C}{D}\). They are fundamental when dealing with equations.

• In algebra, manipulating rational expressions often involves finding a way to express the equation in a simplified form.
• When working with rational expressions, pay close attention to operations involving addition, subtraction, multiplication, and division as they follow specific algebraic rules.
• Take the reciprocal of both sides in an equation as a strategy to isolate or manipulate these expressions in your favor, as we did to solve for "D."

Mastering rational expressions allows you to become adept at transforming complicated equations into simpler forms, making it possible to isolate and solve for specific variables.

Simplifying expressions

Simplifying expressions is a crucial part of solving equations. It involves reducing an expression to its simplest form, often by combining like terms or reducing fractions.

• In our exercise, simplifying comes into play after finding the expression for "D." We started with \(D = \frac{C}{(\frac{A + E}{B})^3}\).
• By multiplying the numerator by \(B^3\) and expanding the denominator as \((A + E)^3\), we simplify the whole expression into a more workable form: \(D = \frac{CB^3}{(A + E)^3}\).
• Fully simplifying an expression makes it easier to substitute values or to understand the relationship between variables at a glance.

Ultimately, simplifying is about making expressions as neat and clear as possible, which reduces errors in calculation and helps clarify the final relationship between variables in a solution.