Question

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

Answer 1

`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)`