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)))#

There isn't a formula—that is, a commonly used method—for solving these kinds of issues, but mathematics is full of surprises, so that doesn't mean a formula can't exist.

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#
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Answer 2

#27#

#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#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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