How do you simplify the fifth root of t (or #t^(1/5)#) multiplied by radical (#16t^5#)?

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Answer 1 · 2015-03-22T17:50:40

I assume that "radical" here means square root.

#t^(1/5)*sqrt(16t^5)=t^(1/5)sqrt16sqrtt^5=4t^(1/5)sqrt(t^5)#

Now as a general definition: #root(n)(x^m)=x^(m/n)#. So #sqrt(t^5)=t^(5/2)#

The expression now becomes:

#t^(1/5)*sqrt(16t^5)=4t^(1/5)sqrt(t^5)=4t^(1/5)t^(5/2)#

An important property of exponents assures us that:
#x^nx^m=x^(n+m)#.

So we can simplify #t^(1/5)t^(5/2)# by adding #1/5+5/2=27/10#.

#t^(1/5)*sqrt(16t^5)=4t^(1/5)t^(5/2)=4t^(27/10)=4 root(10)(t^27)#.