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
Step 1: Write 45^ in the form (1)/(sqrt(a)). We know that 45^ = sqrt(2)2. To express this in the form (1)/(sqrt(a)), we can rationalize the numerator: 45^ = sqrt(2)2 = sqrt(2)sqrt(4) To get 1 in the numerator, we can divide both the numerator and the denominator by sqrt(2): sqrt(2)sqrt(4) = sqrt(2) ÷ sqrt(2)sqrt(4) ÷ sqrt(2) = (1)/(sqrt(4)2) = (1)/(sqrt(2)) So, 45^ = (1)/(sqrt(2)). Comparing this to (1)/(sqrt(a)), we find that a=2. Since a=2 is a positive integer, this form is correct. Step 2: Simplify sqrt(8). We can write sqrt(8) as: sqrt(8) = sqrt(4 × 2) = sqrt(4) × sqrt(2) = 2sqrt(2) Step 3: Substitute the values into the expression sqrt(8)1+ 45^. Using 45^ = (1)/(sqrt(2)) and sqrt(8) = 2sqrt(2): sqrt(8)1+ 45^ = 2sqrt(2)1+(1)/(sqrt(2)) Step 4: Simplify the denominator. 1+(1)/(sqrt(2)) = sqrt(2)sqrt(2) + (1)/(sqrt(2)) = sqrt(2)+1sqrt(2) Step 5: Substitute the simplified denominator back into the main expression. 2sqrt(2)sqrt(2)+1sqrt(2) = 2sqrt(2) × sqrt(2)sqrt(2)+1 Multiply the numerators: 2sqrt(2) × sqrt(2) = 2 × (sqrt(2))^2 = 2 × 2 = 4 So the expression becomes: (4)/(sqrt(2)+1) Step 6: Rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(2)-1: (4)/(sqrt(2)+1) × sqrt(2)-1sqrt(2)-1 = 4(sqrt(2)-1)(sqrt(2))^2 - 1^2 = 4sqrt(2)-42-1 = 4sqrt(2)-41 = 4sqrt(2)-4 The simplified expression in surd form is: 4sqrt(2)-4 That's 2 down. 3 left today — send the next one.