Question

Is the product of 2 irrational numbers always irrational?



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Answer 1

Answer:

No, it is not necessary. For example, √2 × √18 =√36 =6.

Answer 2

Product of two irrational numbers

First of all let us discuss what are rational and irrational numbers.

• Rational Numbers :- Numbers which can be represented in the form of p/q where p and q are integers and q is not equal to 0 are known as rational numbers. Eg. :-  0, 2, 4/5, 10, 13/7 etc.

• Irrational Numbers :- Numbers which cannot be represented in the form of p/q are known as irrational numbers. Eg. :- √2, √5, √7 , pi($\pi$) etc.

Now, back to your question.

For finding the answer to it, let us see some examples :-

• √2 x √2 = 2  ( product is rational)
• √2 x √8 = √16 = 4  (product is rational)
• √5 x √5 = 5  (product is rational)
• √5 x √20 = √100 = 10  (product is rational)
• √2 x √5 = √10 (product is irrational)
• √7 x √3 = √21  (product is irrational)
• √5 x √7 = √35  (product is irrational)
• √10 x √3 = √30  (product is irrational)

Now, in the above examples we can see that product of two irrational numbers is rational in the first four cases whereas it is irrational in the next four cases.

∴ Product of two irrational numbers is not always irrational.