Question

Sqrt(97+56*sqrt(3))/sqrt(7+4*sqrt(3))*sqrt(7-4sqrt(3))

Answer 1

$\frac{\sqrt{97+56\sqrt{3} } }{\sqrt{7+4\sqrt{3} }*\sqrt{7-4\sqrt{3} }} =\frac{\sqrt{49+56\sqrt{3}+48 } }{\sqrt{7+4\sqrt{3} }*\sqrt{7-4\sqrt{3}}}=\frac{\sqrt{(7+4\sqrt{3})^{2}} }{\sqrt{(7+4\sqrt{3})(7-4\sqrt{3})}}=\frac{7+4\sqrt{3} }{\sqrt{49-48} } =7+4\sqrt{3}$

Answer 2

We can simplify the expression by rationalizing the denominator.

First, let's simplify the expression inside the square root in the numerator:

√(97 + 56√3) = √(163 + 81 + 56√3) = √(163 + 9^2 + 294√3 + (4√3)^2) = √((9 + 4√3)^2) = 9 + 4√3

Similarly, let's simplify the expression inside the square root in the denominator:

√(7 + 4√3) = √(3 + 4 + 4√3) = √[(√3)^2 + 2(√3)(2) + 2^2] = √(2 + √3)^2 = 2 + √3

So, the expression becomes:

(9 + 4√3)/(2 + √3)(√(7 - 4√3))

To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator:

(9 + 4√3)/(2 + √3)(√(7 - 4√3)) * (√(7 + 4√3))/(√(7 + 4√3))

= (9 + 4√3)(√(7 + 4√3))/((2 + √3)(√(7 + 4√3))(√(7 - 4√3)))

= (9 + 4√3)(√(7 + 4√3))/((2 + √3)(√(49 - 48)))

= (9 + 4√3)(√(7 + 4√3))/((2 + √3)(1))

= (9 + 4√3)(√(7 + 4√3))/(2 + √3)

Therefore, the simplified expression is:

(9 + 4√3)(√(7 + 4√3))/(2 + √3)