How do you simplify #\frac { a ^ { \frac { 1} { 5} } a ^ { \frac { 6} { 5} } } { a ^ { \frac { 9} { 3} } }#?

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Answer 1 · 2017-07-04T23:34:46

#root(5)(a^8)#

Since the base of the exponents are all #a#, we can apply the exponent product rule in which #a^ba^c=a^(b+c#

With that, we will get

#a^(1/5+6/5)/a^(9/3)#

We can add the fractions of the exponent in the numerator to get:

#a^(7/5)/a^(9/3)#

Similar to the exponent product rule, we can use the divisibility rule which states that #a^b/a^c=a^(b-c)#

Using that, we will simplify the equation to

#a^(7/5-9/3)#

We can also now reduce the fraction #9/3# to just #3# as they are equal.

New expression: #a^(7/5-3)#

Just like subtracting regular fractions, we need to change the denominator of both fractions to a common multiple, in this case, it is #5#.

#a^(7/5-3)=a^((7/5)-3/1*5/5#

#a^(7/5-3)=a^((7/5)-15/5#

Subtracting the exponents gets us #a^(-8/5)#

Now, using the negative exponent rule, we know that #a^(-x)=1/a^x#

This will get us #1/a^(8/5#

We now have to use the fractional exponent law: #a^(x/y)=root(y)(a^x)#

Finally, since #8=x# and #y=5#, we have the simplifed version of #a#:

#1/root(5)(a^8)#