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1.

Simplify :`((25)^(3/2)x(243)^(3/5))/((16)^(5/4)x(8)^(4/3))`

Answer» `((25)^(3/2)x(243)^(3/5))/((16)^(5/4)x(8)^(4/3))`
`((5^2)^(3/2)*(3^5)^(3/5))/((2^4)^(5/4)*(2^3)^(4/3))`
`(5^3*3^3)/(2^5*2^4)`
`(5^3*3^3)/2^9`
`((5*3)/(2^3))^3`
`(15/8)^3`.
Discussion
2.

Assuming that `x , y` are positive real numbers, simplify each of the following: (i)`sqrt(x^(-2)y^3)` (ii) `(x^(-2)doty^(-1/2))^2` (iii)`(sqrt(x^(-3)))^5`

Answer» 1)`(sqrt(x^(-2)y^3))^(1/2)`
`=x^(-2/2)*y^(3/2)`
`=y^(3/2)/x`
`=(sqrty)^3/x`
2)`(x^(-2)*y^(-1/2))^2`
`=(x^((-2)*2)*y^((-1/2)*(2)))`
`=1/(ysqrtx^2)`
3)`(sqrtx^(-3))^5`
`(x^(-3+2))^5`
`x^(5-1)`
`x^4`.
Discussion
3.

Show that :`(x^(a(b-c)))/(x^(b(a-c)))*(x^b)/(x^a))^c=1``((x^(a+b))^2(x^(b+c))(x^(c+a))^2)/((x^a x^b x^c)^4)=1`

Answer» `=(x^(a(b-c)))/(x^(b(a-c)))-:(x^b/x^a)^c`
`=(x^(ab-ac)/x^(ab-bc))-:x^(bc)/x^(ac)`
`=x^(ab-ac-(ab-bc))*x^(ac-bc)`
`=x^(ab-ac-ab+bc+ac-bc)`
`=x^0`
`=1`.
Discussion