Simplify the expressions Write the results using only positi

Simplify the expressions. Write the results using only positive exponents. t^5/t^3 (3x^3y)^4 (a^2b^-3)^2/a^3b^2 (x^3y^2)(xy^3) Perform the operation and simplify. Assume all variables are positive. (a) (4)^-3/2 = (b) 3 squareroot 8x^3 y^2 = (c) squareroot 9y^3/z^2 = (d) 8 squareroot 2 - 3 squareroot 8 = Rationalize the denominator 3/squareroot x - squareroot 2. Perform the operations of the complex numbers. Write the answers in standard form of a complex number a + bi. (a) (6 + 5i) - (3 - i) = (b) (5 - 8t)^2 = (c) 3 + 8i/1 + 3i =

Solution

3(a) t⁵/t³ = t⁵*t^-3 = t^5-3 = t² ( as 1/aⁿ= a^-n and a^m *aⁿ= a^m+n)

(b) (3x³y)⁴ = 3⁴(x³)⁴y⁴ = 81x^3*4y⁴= = 81x¹²y⁴ ( as (ab)ⁿ= aⁿbⁿ and (a^m)ⁿ= a^mn).

(c) (a²b^-3)²/a³b² = a^2*2 b^-3*2/ a³b² = a⁴*a^-3/b⁶*b²= a^4-3/b⁶⁺² = a/b⁸ ( as x^-n= 1/xⁿ).

(d) (x³y²)(xy³)= (x³*x)(y² *y³ )= x³⁺¹y²⁺³ = x⁴y⁵.

4.(a) (4)^-3/2 = (2²)^-3/2 = 2^2*(-3/2) = 2^-3 = ½³ = 1/8.

(b) (8x³y²)^1/3 = (2³x³y²)^1/3 = (2³)^1/3(x³)^1/3 (y²)^1/3= 2^3*1/3 x^3*1/3y^2*1/3 = 2xy^2/3.

(c ) (9y³/z²) = (3²y³/z²)^1/2 = (3²)^1/2(y³)^1/2/(z²)^1/2= 3^2*1/2y^3*1/2/z^2*1/2 = 3y^3/2/z ( as (a/b)ⁿ= aⁿ/bⁿ).

   (d) 82 -38 = 82 -3(4*2) = 82 -32²*2 = 82 -3*22 = 82 -62 = 22 = 2*2^1/2 = 2^3/2.

5. 3/(x-2) = 3(x+2)/ (x-2) (x+2) = 3(x+2)/[(x)²-(2)²]= 3(x+2)/(x-2) ( as (a+b)(a-b) = a² –b²).

6. (a)(6+5i)-(3-i) = 6-3+5i+i = 3+6i.

    (b) (5-8i)² = 5²-2*5*8i +(8i)² = 25-80i +8²i² =25-80i -64i = -39-80i ( as (a-b)² = a²-2ab+b² and i² = -1).

   (c) (3+8i)/(1+3i) = (3+8i)(1-3i)/(1+3i)(1-3i) = ( 3+8i-9i-24i²)/[1²-(3i)²] = ( 27-i)/10 = (27/10)-i/10 ( as i²=-1).