Express the numerical coefficient as a power of its prime factors.

Here are the solutions to the problems. a) Simplify [5]15625x^5y^8c^10 Step 1: Express the numerical coefficient as a power of its prime factors. We find that 15625 = 5^6. So, the expression becomes [5]5^6 x^5 y^8 c^10. Step 2: Apply the fifth root to each term using the property [n]a^m = a^m/n. [5]5^6 x^5 y^8 c^10 = 5^6/5 x^5/5 y^8/5 c^10/5 = 5^1 + 1/5 x^1 y^1 + 3/5 c^2 = 5 · 5^1/5 · x · y · y^3/5 · c^2 = 5xyc^2 [5]5y^3 The simplified expression is 5xyc^2 [5]5y^3. b) Let R be a universal set and P and Q are subsets of R where, n(P) = 50, n(Q) = 30, n(P Q)' = 30 and n(P Q) = 10, what is the value of (i) n(P Q) (ii) n(R) b) (i) Find n(P Q) Step 1: Use the formula for the number of elements in the union of two sets. n(P Q) = n(P) + n(Q) - n(P Q) Substitute the given values: n(P Q) = 50 + 30 - 10 n(P Q) = 80 - 10 n(P Q) = 70 The value of n(P Q) is 70. b) (ii) Find n(R) Step 1: Use the complement rule for sets. The number of elements in the complement of a set A in a universal set U is n(A') = n(U) - n(A). Here, U = R and A = P Q. n(P Q)' = n(R) - n(P Q) Substitute the given values: 30 = n(R) - 10 n(R) = 30 + 10 n(R) = 40 The value of n(R) is 40. c) If and are the roots of the equation 2x^2 - 3x + 4 = 0. Find the value of (i) (1)/() + (1)/() (ii) (( - )^2)/( ) Step 1: Identify the coefficients of the quadratic equation ax^2 + bx + c = 0. For 2x^2 - 3x + 4 = 0, we have a=2, b=-3, c=4. Step 2: Calculate the sum and product of the roots. Sum of roots: + = -(b)/(a) = -((-3))/(2) = (3)/(2) Product of roots: = (c)/(a) = (4)/(2) = 2 c) (i) Find (1)/() + (1)/() Step 3: Simplify the expression and substitute the values of the sum and product of roots. (1)/() + (1)/() = ( + )/( ) = (3/2)/(2) = (3)/(2) × (1)/(2) = (3)/(4) The value of (1)/() + (1)/() is (3)/(4). c) (ii) Find (( - )^2)/( ) Step 4: Use the identity ( - )^2 = ( + )^2 - 4 and substitute the values. (( - )^2)/( ) = (( + )^2 - 4 )/( ) Substitute + = (3)/(2) and = 2: = ((3)/(2))^2 - 4(2)2 = (9)/(4) - 82 = (9)/(4) - (32)/(4)2 = (-23)/(4)2 = -(23)/(4) × (1)/(2) = -(23)/(8) The value of (( - )^2)/( ) is -(23)/(8). Send me the next one 📸