Problem 51 - College Algebra

Question

(Hint: To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \((\sqrt[n]{y})^{n}=y .\) ) $$ f(x)=\sqrt[3]{x}+1 $$

Step-by-Step Solution

Answer

The isolated variable x can be written as \(x = (f(x)-1)^3\)

Step 1: Identify the Variable inside the Root

The variable we're trying to isolate is \(x\) and it's placed within a cube root in the function \(f(x)=\sqrt[3]{x} + 1\). So the primary task involves eliminating the cube root.

Step 2: Move Constants

In order to do that, the constant '1' needs to be moved to the other side of the equation. We can perform this operation by subtracting '1' from both sides of the equation, which gives us: \(f(x) - 1 = \sqrt[3]{x}\)

Step 3: Cube Both Sides

Now, we cube both sides of the equation to remove the cube root from around x. This gives us: \((f(x) - 1)^3 = x\)

Key Concepts

Cube Root

The cube root is a special type of root that reverses the effect of cubing a number. It's different from square roots because instead of needing a number that multiplied by itself gives the original number, the cube root of a number is the value that, when multiplied by itself twice more (cubed), returns the original number. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
Cube roots can be written in several ways:

• The radical notation: \(\sqrt[3]{x}\), where the little 3 is crucial, as it distinguishes cube roots from square roots.
• Exponent form: \(x^{\frac{1}{3}}\), which is particularly useful in algebraic transformations.

The cube root has a unique property: unlike square roots, every real number has exactly one real cube root, which makes it quite straightforward to work with in equations. In the context of the provided exercise, eliminating the cube root helps isolate the variable \(x\). This is achieved by cubing both sides of the equation as outlined in the step-by-step solution. This inverse operation effectively simplifies the problem to solve for \(x\).
Understanding the concept of cube roots is essential for tackling algebraic problems where variables are embedded under a cubic radical.

Exponential Equations

Exponential equations are equations where the variables appear as exponents. They are common in various fields of science and mathematics, often used to describe growth patterns.
Solving these types of equations often involves using logarithms or transforming the base of the exponent to isolate the variable. In the exercise provided, even though the emphasis is on roots, the idea of exponentiation is still crucial. By cubing both sides of the equation, you're essentially applying an exponential operation (raising both sides to the power of 3).
Key points to remember in exponential equations:

• To "undo" an exponent (like a cube root), you apply the inverse operation, such as cubing.
• Balancing both sides of an equation is essential – what you do to one side, you must do to the other.
• Understanding properties of exponents can drastically simplify solving these equations.

In the case of the exercise, the exponential move made it possible to eliminate the cube root and simplify the equation to a standard algebraic equation, which is more straightforward to solve for \(x\). This showcases the versatility of exponential transformations in solving mathematical problems.

Algebraic Functions

Algebraic functions are the building blocks of algebra. They are equations that involve polynomials, roots, and rational numbers. In simple terms, an algebraic function expresses a relationship between variables using algebraic operations such as addition, subtraction, multiplication, division, and taking roots.
In the given exercise, the function \(f(x)=\sqrt[3]{x} + 1\) is an algebraic function because it includes a cube root, an operation that perfectly fits within the umbrella of algebraic manipulations.

• Understanding algebraic functions is crucial because they often form the backbone of more complex functions.
• Simplifying algebraic functions often involves removing radicals, as seen in the exercise, to isolate and solve for variables.
• Algebraic functions can have many forms, from simple linear functions to complicated polynomial expressions.

By manipulating these types of functions using operations like isolating the variable and addressing roots, students can solve for unknowns and explore the relationships between different variables. This exercise illustrates one of the fundamental aspects of working with algebraic functions by elucidating methods to deal with cube roots, which are a common feature in these equations.