Question

Simplify the following:

`(27/343)^(2/3) ÷ (1)/(625/1296)^(1/4) xx (536)/root(3)(27)`

Answer 1

`(27/343)^(2/3) ÷ (1)/(625/1296)^(1/4) xx (536)/root(3)(27)`

= `(3^3/7^3)^(2/3) ÷ (1)/(((5^4)/(2^4 xx 3^4))^(1/4)) xx (2^3 xx 67)/root(3)(3^3)`

= `(3^3/7^3)^(2/3) ÷ (1)/(((5^4)/(2^4 xx 3^4))^(1/4)) xx (2^3 xx 67)/(3^3)^(1/3)`

= `(3^(3xx2/3)/(7^(3xx2/3))) ÷ (1)/((5^(4xx1/4)/(2^(4xx1/4) xx 3^4xx1/4))) xx (2^3 xx 67)/(3^(3xx1/3)`  .....(Using (a^m)ⁿ = a^mn)

= `(3^2/7^2) ÷ (1)/((5^1/(2^1 xx 3^1))) xx (2^3 xx 67)/(3^1)`

= `(3^2/7^2) ÷ ((2^1 xx 3^1)/5^1) xx ((2^3 xx 67)/(3^1))`   ......(Using a^m x aⁿ = a^m+n and a^m ÷ aⁿ = a^m-n)

= `(3^2/7^2) ÷ (5^1/(2^1 xx 3^1)) xx ((2^3 xx 67)/(3^1))`

= `3^(2-1-1) xx 2^(3-1) xx 5^1 xx 7^2 xx 67`
= 3⁰ x 2² x 5¹ x 7² x 67
= 1 x 4 x 5 x 49 x 67  ......(Using a⁰ = 1)
= 65660.