This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To evaluate the expression sqrt(2)sqrt(2)-sqrt(3) - sqrt(3)sqrt(3)+sqrt(2) by rationalizing the denominator, we will rationalize each fraction separately. Step 1: Rationalize the first term, sqrt(2)sqrt(2)-sqrt(3). Multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(2)+sqrt(3). sqrt(2)sqrt(2)-sqrt(3) × sqrt(2)+sqrt(3)sqrt(2)+sqrt(3) = sqrt(2)(sqrt(2)+sqrt(3))(sqrt(2))^2 - (sqrt(3))^2 = 2 + sqrt(6)2 - 3 = 2 + sqrt(6)-1 = -2 - sqrt(6) Step 2: Rationalize the second term, sqrt(3)sqrt(3)+sqrt(2). Multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(3)-sqrt(2). sqrt(3)sqrt(3)+sqrt(2) × sqrt(3)-sqrt(2)sqrt(3)-sqrt(2) = sqrt(3)(sqrt(3)-sqrt(2))(sqrt(3))^2 - (sqrt(2))^2 = 3 - sqrt(6)3 - 2 = 3 - sqrt(6)1 = 3 - sqrt(6) Step 3: Subtract the rationalized second term from the rationalized first term. (-2 - sqrt(6)) - (3 - sqrt(6)) = -2 - sqrt(6) - 3 + sqrt(6) Step 4: Combine like terms. (-2 - 3) + (-sqrt(6) + sqrt(6)) = -5 + 0 = -5 The final answer is -5. That's 2 down. 3 left today — send the next one.