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
Here are the solutions for the given problems: a) Simplify (15u^3v^2)/(7st^3) × (28s^3t^2)/(5u^3v) Step 1: Combine the numerators and denominators. (15u^3v^2 × 28s^3t^2)/(7st^3 × 5u^3v) Step 2: Multiply the numerical coefficients and group like variables. ((15 × 28) × u^3 × v^2 × s^3 × t^2)/((7 × 5) × u^3 × v × s × t^3) Step 3: Simplify the numerical coefficients. (420)/(35) = 12 Step 4: Simplify the variables using the exponent rule (x^a)/(x^b) = x^a-b. For u: (u^3)/(u^3) = u^3-3 = u^0 = 1 For v: (v^2)/(v^1) = v^2-1 = v^1 = v For s: (s^3)/(s^1) = s^3-1 = s^2 For t: (t^2)/(t^3) = t^2-3 = t^-1 = (1)/(t) Step 5: Combine all simplified parts. 12 × 1 × v × s^2 × (1)/(t) = (12s^2v)/(t) The simplified expression is (12s^2v)/(t). b) In a geometric progression, the third term is (2)/(9). Let the first term of the geometric progression be a and the common ratio be r. The formula for the n-th term of a geometric progression is T_n = ar^n-1. For the third term (n=3), we have: T_3 = ar^3-1 = ar^2 Given that the third term is (2)/(9): ar^2 = (2)/(9) This equation relates the first term a and the common ratio r of the geometric progression.