Irrational Numbers

To solve this problem, we need to find the value of that satisfies the equation:

1. Express Both Sides with the Same Base:
We know that , so we can rewrite as:

2. Equating Powers of the Same Base:
Now we equate the exponents since the bases are the same:

Final Answer:
The correct option is (C) 6.

To solve the problem, we need to classify the nature of the number .

1. Understanding the Nature of :
The number (pi) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. Its approximate value is 3.14159, and it continues infinitely without repeating.

2. Rational vs Irrational Numbers:
- A rational number can be expressed as a fraction where and are integers and .
- An irrational number cannot be expressed as such a fraction. Its decimal representation is non-terminating and non-repeating.

3. Properties of :
Since cannot be expressed as a ratio of two integers and has a non-terminating, non-repeating decimal expansion, it is classified as an irrational number.

Final Answer:
The number is an irrational number.

Step 1: Understand what a prime number is.

A prime number is a number that has exactly two distinct positive divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, etc.

Step 2: Consider the square root of a prime number.

Let’s take examples:

• — not a rational number
• — not a rational number
• — not a rational number

Step 3: Can the square root of a prime number be rational?

Suppose is rational, where is a prime.

Then it can be expressed as: where and are integers, , and .

Squaring both sides: This implies that is divisible by , hence is divisible by . So for some integer .

Substituting back: Now is divisible by , which implies is divisible by . But that contradicts our assumption that and have no common factor.

Step 4: So, is irrational when is a prime number.

The correct option is (D): Irrational number