Simplify the following expressions a 2x12 y4xz2 b log 2x 3

Simplify the following expressions (a) (2x^1/2 y^4/xz)^-2 (b) log 2x - 3 log(y) (c) x^2 3 squareroot 27 y^5/(squareroot x)^3 2y^-2

Solution

(a) [(2x^1/2y⁴)/(xz)]^-2 = (2x^1/2-1 y⁴z^-1)^-2 = (2x^-1/2 y⁴z^-1)^-2 = 2^-2 (x^-1/2 )^-2 (y⁴)^-2(z^-1)^-2 =2^-2 xy^-8z² = xz²/4y⁸.

          [ as (a^m)ⁿ = a^mn, a^-m= 1/a^m , 1/a^-m = a^m, a^m.aⁿ = a^m+n and (abc)^m = a^m b^m c^m].

(b) log 2x -3log (y) = log 2x -log (y³) log(2x/y³) ( as log a^m = mlog a and log (m/n) = log m –log n).

(c) [x²(27y⁵)^1/3]/[ (x)³ 2y^-2] = [x²(3³ y⁵)^1/3]/[ (x^1/2)³2y^-2] =[(x²)*3*y^5/3]/[ 2x^3/2y^-2] = (3/2)[ (x^2-3/2)y^5/3+2]= (3/2)(x^1/2y^11/2) = (3/2)(xy¹¹).