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Prove this is irrational 10√7 + 7√3

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Answer:

Proof is given below, don't worry ;)

Step-by-step explanation:

On the contrary, LET us ASSUME that 10√7 + 7√3 is RATIONAL

For the sum of two numbers to be a rational number, both  the numbers have to be rational numbers, which means 10√7 and 7√3 are both rational numbers,

or 10√7 and 7√3 can both be represented in the form of p/q, where p and q are co-prime integers, and q is not EQUAL to zero.

Let us assume that 7√3=p/q

Then √3=p/7q

p and 7q are both rationals, but √3 is irrational. Thus, there is an inequality.

The inequality has occurred because our ASSUMPTION that 7√3 is rational is wrong.

Similarly, it can be proved that 10√7 is irrational as well.

Thus, sum of two irrationals is always an irrational number.

So, 10√7 + 7√3 is irrational.

∴ proved.

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