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
Alright wild — let's do this. Here are the solutions for questions 6 and 7. 6. Rationalize the denominator: (2)/(3sqrt(2)-5) Step 1: To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 3sqrt(2)-5 is 3sqrt(2)+5. (2)/(3sqrt(2)-5) × 3sqrt(2)+53sqrt(2)+5 Step 2: Multiply the numerators and the denominators. 2(3sqrt(2)+5)(3sqrt(2)-5)(3sqrt(2)+5) Step 3: Expand the numerator and use the difference of squares formula (a-b)(a+b) = a^2 - b^2 for the denominator. 2 · 3sqrt(2) + 2 · 5(3sqrt(2))^2 - (5)^2 6sqrt(2) + 109 · 2 - 25 6sqrt(2) + 1018 - 25 Step 4: Simplify the denominator. 6sqrt(2) + 10-7 This can also be written by moving the negative sign to the front or distributing it to the numerator. -6sqrt(2) + 107 7. Simplify: 14sqrt(20) - 8sqrt(20) - 10sqrt(5) Step 1: Simplify the surd sqrt(20) by finding its prime factors. sqrt(20) = sqrt(4 × 5) = sqrt(4) × sqrt(5) = 2sqrt(5) Step 2: Substitute the simplified surd back into the expression. 14(2sqrt(5)) - 8(2sqrt(5)) - 10sqrt(5) Step 3: Perform the multiplications. 28sqrt(5) - 16sqrt(5) - 10sqrt(5) Step 4: Combine the like terms (terms with sqrt(5)). (28 - 16 - 10)sqrt(5) (12 - 10)sqrt(5) 2sqrt(5) Send me the next one 📸