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

2 Answers
Jun 6, 2017

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

Jun 7, 2017

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