Problem 110 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(n \in N\) such that \((7+4 \sqrt{3})^{\mathrm{n}}=I+f\), where \(I \in N\) and \(0

Step-by-Step Solution

Answer

The value of \((I+f)(I-f)\) is 1.

Step 1: Understand the Problem Statement

Given the expression \((7+4\sqrt{3})^{n} = I + f\) where \(I\) is an integer and \(0 < f < 1\). We need to find the value of \((I + f)(I - f)\). The expression is a consequence of the powers of a surd and is closest to an integer.

Step 2: Analyze the Conjugate Expression

Consider the conjugate expression of \((7 + 4\sqrt{3})^n\), which is \((7 - 4\sqrt{3})^n\). For large \(n\), \((7 - 4\sqrt{3})^n\) becomes very small because \(7 - 4\sqrt{3} < 1\) and raising a number less than one to a power results in a smaller number approaching 0.

Step 3: Sum and Difference of Conjugates

Note that \((7+4\sqrt{3})^n + (7-4\sqrt{3})^n\) will also be an integer. Since \((7 - 4\sqrt{3})^n\) is very small, this sum approximates \((7+4\sqrt{3})^n = I + f\), making it close to \(I\).

Step 4: Product of Conjugate Expressions

Compute the product \((7+4\sqrt{3})^n \cdot (7-4\sqrt{3})^n\). The product of these conjugates will be \([(7)^2 - (4 \sqrt{3})^2]^n = (49 - 48)^n = 1^n = 1\).

Step 5: Calculate \((I + f)(I - f)\)

The expression \((I - f)\) can be taken to be approximately \((7-4\sqrt{3})^n\), which is very small. Thus, \((I+f)(I-f) = ((7+4\sqrt{3})^n)((7-4\sqrt{3})^n) = 1\).

Step 6: Conclusion to the Solution

Since \((I + f)(I - f) = 1\), the answer to the problem is that the value is indeed 1.

Key Concepts

Powers of Surds

Surds are expressions that contain roots, such as square roots, that cannot be simplified into a rational number. Understanding the powers of surds is important when dealing with expressions like \( (7 + 4\sqrt{3})^n \). As you increase the powers of surds, they become either very large or very small numbers depending on the values raised to those powers. When a number, particularly a surd, is raised to a power, the nature of that number dictates how the resulting expression behaves:

• If the base of the surd is greater than 1, increasing powers will result in large numbers.
• If the base is less than 1, increasing powers will result in numbers that shrink, eventually getting closer to zero.

In the given problem, the expression \( (7 + 4\sqrt{3})^{ n } \), where \( (7 + 4\sqrt{3}) \) is greater than 1, will be a large number but very close to an integer form. This is due to its partner, the conjugate expression, contributing a negligible amount when summed.

Conjugate Expressions

Conjugate expressions play a significant role in simplifying and solving mathematical problems, especially those involving roots or surds. A conjugate is formed by changing the sign between two terms in a binomial expression. For instance, if we have \( 7 + 4\sqrt{3} \), its conjugate is \( 7 - 4\sqrt{3} \).Conjugates are powerful tools because:

• Their product results in an integer when surds are involved. This stems from the identity \( (a+b)(a-b) = a^2 - b^2 \), which in our case simplifies to an integer.
• They help to rationalize denominators or to simplify complex radical expressions.

In the problem, by using the conjugates \( (7 + 4\sqrt{3})^n \) and \( (7 - 4\sqrt{3})^n \), we derive that the product of these two terms is 1, showing that even though each term is large or small individually, their product simplifies nicely.

Integer Approximation

In many mathematical contexts, particularly those involving powers and surds, the idea of integer approximation comes into play. Integer approximation refers to how a number, which may not be whole, can be approximated or rounded to the nearest whole number.The expression \( (7 + 4\sqrt{3})^n = I + f \) uses this concept. Here:

• \( I \) is an integer that approximates the expression value closely.
• \( f \) is a fractional part that is very small (\( 0 < f < 1 \)).

This concept shows how near-perfect expressions can behave like integers when their fractional parts are minimal relative to their integers—especially for large powers. In the problem, even though \( (7+4\sqrt{3})^n \) is not exactly an integer, it is exceptionally close, leading to a product \( (I + f)(I - f) \) that resolves neatly with the help of its conjugate counterpart.