Let us first begin by labelling the circles to make workings easier. The top circle is x, the left circle is y, and the right circle is z. Because the surd expressions are products, it is clear that multiplication/division is needed to solve the problem. There was nothing to divide by, so multiplying them together is the only option. The only relevant two are:
x²yz = -28√2 - 14
x²y²z² = 196
Because the last one can be rephrased into (xyz)² and it is a square number, square rooting it gives the integer 14, which is also xyz. x²yz/xyz = x, and as both of these variables are available, we can calculate the value of x.
(-28√2 - 14)/14 = -2√2 - 1 = x.
Of course xy/x and xz/x are y and z respectively, so calculating those is the last step of the proof.
(-6 + 2√2) /(-2√2 - 1) = y
Rationalising the denominator...
((-6 + 2√2)(2√2 + 1))/((-2√2 - 1)(2√2 + 1))
Expanding the brackets...
(6 + 8 - 12√2 - 2√2)/(1 - 8 - 2√2 + 2√2)
Simplifying...
(14 - 14√2)/-7
And dividing.
-2 + 2√2 = y
The same goes for xz/z, but I will remove the step by step explanation this time.
(7 + 7√2)/(-2√2 - 1)
((7 + 7√2)(2√2 + 1))/((-2√2 - 1)(2√2 + 1))
(7 + 28 + 14√2 + 7√2)/(1 - 8 - 2√2 + 2√2)
(35 + 21√2)/-7
-5 - 3√2 = z
And these are the full answers.
Top circle: -2√2 - 1
Left circle: 2√2 - 2
Right circle: -3√2 - 5