# How to solve square roots with exponents on the outside?

To simplify square roots with exponents on the outside, or radicals, apply the rule nth root of a^n = a. A radical in the form `root(n)(x)` can be simplified using the radical rule:

`root(n)(a^n) = a`

To apply this rule, consider this example.

Example 1: What is the simplified form of `root(3)(x^12)` ?

The index of this radical is n=3. Since the index is 3, express the x^12 with the factor x^3.

`root(3)(x^12)`

`=root(3)(x^3*x^3*x^3*x^3)`

Then, apply the radical rule `root(n)(a*b) = root(n)(a) * root(n)(b)` .

`=root(3)(x^3)*root(3)(x^3)*root(3)(x^3)*root(3)(x^3)`

Apply the radical rule `root(n)(a^n) = a` .

`=x * x * x * x`

`=x^4`

Therefore, the given radical simplifies to `root(3)(x^12) = x^4` .

Example 2: Simplify `root(5)(y^8z^7)` .

The index of the radical is n=5. So factor the variables in such a way that their factors contain exponent 5.

`root(5)(y^8z^7)`

`=root(5)(y^5*y^3 *z^5*z^2)`

Then, apply the radical rule `root(n)(a * b) =root(n)(a) * root(n)(b) .`

`=root(5)(y^5)*root(5)(y^3)*root(5)(z^5)*root(5)(z^2)`

Apply the radical rule `root(n)(a^n)=a.`

`=y*root(5)(y^3) * z*root(5)(z^2)`

`=yz*root(5)(y^3)*root(5)(z^2)`

Since the factors y^3 and z^2 have exponents less than the index, they remain inside the radical sign. To multiply these two radicals, apply the rule: `root(n)(a)*root(n)(b) = root(n)(a*b).`

`=yz root(5)(y^3z^2)`

Therefore, the given radical simplify to

`root(n)(y^8z^7) = yz root(5)(y^3z^2). `

Example 3: What is the simplified form of `root(4)(288)? `

To simplify, express 288 with its prime factorization.

`root(4)(288)`

`=root(4)(2*2*2*2*2*3*3*)`

The index of the radical is n=4. Group same factors in such a way that it will have exponent 4.

`=root(4)(2^4*2 *3^2)`

Apply the radical rule `root(n)(a*b)=root(n)(a)*root(n)(b).`

`=root(4)(2^4)*root(4)(2)*root(4)(3^2)`

Then, apply `root(n)(a^n)=a` .

`=2*root(4)(2)*root(4)(3^2)`

Since the factors 2 and 3^2 have exponents less than the index, they remain inside the radical sign.

`=2root(4)(2*3^2)`

`=2root(4)(18)`

Therefore, it simplifies to `root(4)(288)=2root(4)(18)` .

Approved by eNotes Editorial Team