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

1 Answer

root(5)(a^8)

Explanation:

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)