is square root of rpi squared irrational or rational

Question

is 9pi irrational or rational
adding them together is irrational or rational

Answer 1

Let's analyze each part of your question step by step:

1. Square root of π²:

   • The square root of π² is |π| (the absolute value of π).
   • Since π is an irrational number, |π| is also irrational.
   • Therefore, the square root of π² is irrational.

2. 9π:

   • Here, 9 is a rational number and π is an irrational number.
   • The product of a rational number (9) and an irrational number (π) is irrational.
   • Thus, 9π is irrational.

3. Adding them together (√(π²) + 9π):

   • We have |π| (irrational) + 9π (irrational).
   • The sum of two irrational numbers is not guaranteed to be irrational. It can be either rational or irrational depending on the specific numbers involved.
   • In this case, |π| (which equals π) and 9π are both irrational, but their sum (π + 9π) simplifies to 10π.
   • Since 10π is the product of an integer (10) and an irrational number (π), it is irrational.

Conclusion:

• The square root of π² is irrational.
• 9π is irrational.
• Their sum, |π| + 9π (which is 10π), is also irrational.