Determining the irrationality of a quotient

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In summary, we are asked to determine whether the quotient (7)1/2/(15)1/3 is rational or irrational. Using previous theorems, we know that (7)1/2 is irrational because it is a prime, and (15)1/3 can be split into (5)1/3 times (3)1/3. We also know that (5)1/3 is irrational. To prove that the quotient is irrational, we use the uniqueness of prime factorization and assume it is rational in lowest terms. By taking the sixth power of both sides and using arguments about prime divisibility, we can obtain a contradiction, proving that the quotient is indeed irrational.
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trap101
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Determine whether (7)1/2/(15)1/3 is either rational or irrational and prove your answer is correct.

So I know that (7)1/2 is irrational from previous theorems since it is a prime, I also split up (15)1/3 into (5)1/3 times (3)1/3. I previously had shown that (5)1/3 is also irrational. Doing this question I showed that (3)1/3 is irrational by the uniqueness of prime factorization.

But my problem lies in showing that this whole quotient is irrational. I don't know where to start. Maybe assume that the quotient is rational and obtain a contradiction? If so how could I start it?
 
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trap101 said:
Determine whether (7)1/2/(15)1/3 is either rational or irrational and prove your answer is correct.

So I know that (7)1/2 is irrational from previous theorems since it is a prime, I also split up (15)1/3 into (5)1/3 times (3)1/3. I previously had shown that (5)1/3 is also irrational. Doing this question I showed that (3)1/3 is irrational by the uniqueness of prime factorization.

But my problem lies in showing that this whole quotient is irrational. I don't know where to start. Maybe assume that the quotient is rational and obtain a contradiction? If so how could I start it?

Yes, assume it's equal to a rational and assume its in lowest terms. Then take the sixth power of both sides. Then make the usual sort of arguments about prime divisibility.
 

FAQ: Determining the irrationality of a quotient

1. What does it mean for a number to be irrational?

A number is considered irrational if it cannot be expressed as a ratio of two integers (whole numbers) and has an infinite number of decimal places without repeating a pattern. In other words, it cannot be written as a fraction or a terminating or repeating decimal.

2. How do you determine if a quotient is irrational?

To determine if a quotient is irrational, you need to divide the numerator (top number) by the denominator (bottom number) using long division or a calculator. If the result is a decimal that never ends or repeats, then the quotient is irrational.

3. Can a quotient be both rational and irrational?

No, a quotient cannot be both rational and irrational. A number is either one or the other, based on whether it can be expressed as a ratio of two integers or not.

4. What is the difference between an irrational quotient and an irrational number?

An irrational quotient is a result of dividing two numbers, whereas an irrational number is any number that cannot be expressed as a ratio of two integers. An irrational quotient may or may not result in an irrational number.

5. Why is it important to determine the irrationality of a quotient?

Determining the irrationality of a quotient is important because it helps us understand the nature of numbers and their properties. It also helps in solving mathematical problems and in real-life applications, such as in engineering and science. Additionally, knowing if a number is irrational can also help us make informed decisions when dealing with measurements and calculations.

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