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Find the two numbers which on multiplication with √360 gives a rational number. Are these numbers rational or irrational?

User Mandie
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26 votes

Answer:

The two numbers we are looking for are a rational number and an irrational number.

Explanation:

To find the two numbers that, when multiplied by the square root of 360, give a rational number, we can consider the prime factorization of 360. 360 can be written as 2^3 * 3^2 * 5, so the square root of 360 is equal to √(2^3 * 3^2 * 5) = 2√(3 * 5) = 2√15.

Now, suppose we want to find a rational number that is equal to x * √15. We can rewrite this expression as x * √(3 * 5) = x * √3 * √5. For this expression to be rational, both x and √3 must be rational or both must be irrational.

If x is rational, then the expression x * √3 * √5 is rational. Therefore, one of the two numbers we are looking for is x, which is a rational number.

If x is irrational, then the expression x * √3 * √5 is irrational. Therefore, the other number we are looking for is √3, which is an irrational number.

Therefore, the two numbers we are looking for are a rational number and an irrational number.

User Lakerskill
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