Problem 47 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ y=(1+\sqrt[3]{x})^{\frac{3}{3}} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(y = 1+x^{1/3}\).

Step 1: Simplify Cube Root

The cube root of \(x\) is \(x^{1/3}\). Thus, the expression will become \(y = (1+x^{1/3})^3\).

Step 2: Simplify Exponent

We can simplify the exponent, since any number or expression to the power of 3/3 will be itself. Thus, the final expression is \(y = 1+x^{1/3}\).

Key Concepts

Cube Root Simplification

Understanding cube root simplification is essential when solving calculus problems like the one given in the exercise. Essentially, a cube root is the number that, when multiplied by itself three times, returns the original number. In mathematical terms, if we have a number \(x\), then its cube root is expressed as \(\sqrt[3]{x}=x^{1/3}\). This form helps simplify calculations and makes it easier to manipulate expressions, especially when dealing with calculus problems.
To simplify a cube root in an equation:

• Convert \(\sqrt[3]{x}\) to \(x^{1/3}\), using the property that cube root equals raising to the power of one-third.
• This transformation allows us to work with exponents, which are often easier to manipulate.
• Once expressed as an exponent, it opens the door to applying exponent rules.

By expressing cube roots as fractional exponents, we broaden the strategies we can use to handle mathematical operations. This is crucial in higher-level math contexts such as those encountered in IIT JEE Calculus.

Exponent Rules

Exponent rules are fundamental when it comes to simplifying expressions, especially in calculus. Understanding these rules allows us to transform and simplify mathematical expressions more efficiently.
Here are some basic exponent rules:

• \(a^m \times a^n = a^{m+n}\)
• \((a^m)^n = a^{m\cdot n}\)
• \(a^m/a^n = a^{m-n}\)
• \(a^{0} = 1\), given \(a eq 0\)

In the provided problem, using the rule that any expression raised to the power of \(1\) results in the expression itself, we have \((1+x^{1/3})^{3/3} = 1+x^{1/3}\).
This simplification is a powerful tool. Eliminating the exponent simplifies the expression and focuses on the core components of the equation. Recognizing and applying exponent rules is a critical skill for calculus exams such as the IIT JEE.

Expression Simplification

Expression simplification is the process of making a complex expression more manageable and understandable by reducing it to its simplest form. This involves applying operations that refine the expression without changing its value.
Here are steps you might typically take:

• Identify elements that can be simplified, such as square or cube roots.
• Apply exponent rules to remove or combine exponential expressions.
• Consolidate similar terms to further reduce complexity.

For the exercise, simplifying \(y = (1+\sqrt[3]{x})^{3/3}\) resulted in \(y = 1+x^{1/3}\). This reduction helps to better understand the problem by removing unnecessary complexities and focusing on core components.
Expression simplification is especially important in calculus because it can greatly reduce the difficulty of differentiation or integration, aligning with typical questions seen in the IIT JEE exam.