# prove that the difference and quotient of 5 + 2 root 3 and 5 minus 2 root 3 irrational?

### 1 Answer

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- llafferLv 71 month ago
If you mean:

(5 + 2√3) - (5 - 2√3)

Distribute the -1 to get:

5 + 2√3 - 5 + 2√3

4√3

That's irrational.

If dividing:

(5 + 2√3) / (5 - 2√3)

Rationalize the denominator by multiplying both halves by the conjugate of the denominator:

(5 + 2√3)² / [(5 - 2√3)(5 + 2√3)]

(25 + 20√3 + 4 * 3) / (25 - 4 * 3)

(25 + 20√3 + 12) / (25 - 12)

(37 + 20√3) / 13

or:

37/13 + (20/13)√3

And that's still irrational.

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