Given that 2√3+2√3 divided by 2√3-3√2 = m+n√6, find m+n

Answer 1

To simplify the expression, we can use the conjugate of the denominator to eliminate the square roots in the denominator.

The conjugate of 2√3 - 3√2 is 2√3 + 3√2.

To eliminate the square roots in the denominator, we can multiply the numerator and denominator by 2√3 + 3√2:

(2√3 + 2√3)(2√3 + 3√2) / (2√3 - 3√2)(2√3 + 3√2)

Simplifying the numerator:

(2√3 + 2√3)(2√3 + 3√2)
= 4(3) + 2(2√3)(3√2) + 2(2√3)(2√3) + 2(2√3)(3√2)
= 12 + 6√6 + 12 + 12√6
= 24 + 18√6

Simplifying the denominator:

(2√3 - 3√2)(2√3 + 3√2)
= (2√3)^2 - (3√2)^2
= 12 - 18
= -6

Therefore,

(2√3 + 2√3)(2√3 + 3√2) / (2√3 - 3√2)(2√3 + 3√2)
= (24 + 18√6) / -6
= -4 - 3√6

Thus, m = -4 and n = -3, so m + n = -4 + (-3) = -7.

Therefore, m + n = -7.