Question

Is there a formula for `root(x)(a) xx root(y)(a)`? For example, `sqrt(81) xx root4(81)`

Answer 1

`root(x)(a^m)xxroot(y)(a^m)=a^((m(x+y))/(xy))=root(xy)(a^(m(x+y)))`

Explanation:

There is no formula i.e. often used by people to solve such problems. However, mathematics is full of surprises and it does not mean that we cannot have a formula.

Here it is observed that in the example you have square root and fourth roots of `81`, which is itself a power of `3` i.e. `3^4`. Hence we will attempt a formula cosidering `a=b^m` and we attempt

`root(x)(a^m)xxroot(y)(a^m)`

= `(a^m)^(1/x)xx(a^m)^(1/y)`

= `a^(m/x)xxa^(m/y)`

= `a^(m/x+m/y)`

= `a^((m(x+y))/(xy))`

= `root(xy)(a^(m(x+y)))` and that is the formula.

i.e. `root(x)(a^m)xxroot(y)(a^m)=a^((m(x+y))/(xy))=root(xy)(a^(m(x+y)))`

If `a` is not a power than you can use `m=1`

Using this `sqrt(3^4)xxroot(4)(3^4)`

= `3^((4(2+4))/(2xx4))`

= `3^((4xx6)/8)`

= `3^3=27`

Answer 2

`27`

Explanation:

`sqrt81 xx root4(81)`

`:.9 xx root4(3*3*3*3)`

`:.root4(a) xx root4(a) xx root4(a) xx root4(a)=a`

`:.9 xx 3=27`