I understand the concept of multiplying radicals with different indices. I saw the answer but can anyone explain how this one to me? I am having fits trying to figure this one out.

Answer 1

#root(4)(9ab^3)sqrt(3a^4b)=3a^2broot(4)(ab)#

Let us first recall important formulas to be used here. They are

#root(n)a=a^(1/n)# and #root(n)a^m=a^(m/n)# and of course #sqrta=a^(1/2)#
Further #a^pxxa^q=a^((p+q))#.
Therefore #root(4)(9ab^3)sqrt(3a^4b)#
= #(9ab^3)^(1/4)xx(3a^4b)^(1/2)#
= #(3^2ab^3)^(1/4)xx(3a^4b)^(1/2)#
= #3^(2xx1/4)a^(1/4)b^(3xx1/4)xx3^(1/2)a^(4/2)b^(1/2)#
= #3^(2/4)a^(1/4)b^(3/4)xx3^(1/2)a^2b^(1/2)#
= #(3^(1/2)xx3^(1/2))xx(a^(1/4)xxa^2)xx(b^(3/4)xxb^(1/2))#
= #3^1xxa^2xxa^(1/4)xxb^(3/4+1/2)#
= #3a^2a^(1/4)b^(5/4)#
= #3a^2a^(1/4)b^(1+1/4)#
= #3a^2bxx(a^(1/4)b^(1/4))#
= #3a^2broot(4)(ab)#
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Answer 2

Sure, I'd be happy to explain multiplying radicals with different indices.

When you multiply radicals with different indices, you can simplify the expression by first rewriting the radicals with the same index. To do this, you need to find the least common multiple (LCM) of the indices. Once you have the same index for both radicals, you can then multiply the radicands together and simplify if possible.

For example, let's say you have √(2) * ∛(3).

To make the indices the same, you can rewrite √(2) as √(2)^(3/3) and ∛(3) as ∛(3)^(2/2), since 3 is the least common multiple of 2 and 3.

Now you have (√(2)^(3/3)) * (∛(3)^(2/2)), which simplifies to ∛(2^3) * ∛(3^2).

Then, you can multiply the radicands together to get ∛(2^3 * 3^2).

Finally, you can simplify the expression under the radical, which gives you ∛(8 * 9), or ∛(72).

So, √(2) * ∛(3) simplifies to ∛(72).

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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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