Question

How do you simplify `sqrt(4a^6)`?

Answer 1

`sqrt4a^6 = 2a^3`

Explanation:

`sqrt(4a^6) = sqrt4*sqrta^6 = 2a^3`

The square root of `4` is `2`, and to obtain the square root of `a^6` we

need to divide the given exponent by the value of the root required:

So: `sqrta^6=a^(1/2)*a^6=a^(6/2)=a^3`

Answer 2

`2` `a^3`

Explanation:

The expression which we have is:-

`rArrsqrt(4a^6)`

It can also be written as

`rArrsqrt(4a^6)`

`rArrsqrt(2^2xxa^6)`

`rArr sqrt(2^2)xx sqrt(a^6)`

`rArr2xxa^3`

`rArr 2a^3`

The final answer is: `(2a^3)`

Answer 3

`sqrt(4a^6) = abs(2a^3)`

Explanation:

Note that:

`(2a^3)^2 = 2^2(a^3)^2 = 4a^6`

So `2a^3` is a square root of `4a^6`.

Note that:

`(-2a^3)^2 = 4a^6`

So `-2a^3` is a square root of `4a^6` too.

What do we mean when we write `sqrt(4a^6)` ?

So long as the radicand is non-negative, then these symbols refer to the principal non-negative square root.

For any real value of `a`, we have `4a^6 >= 0`, so `sqrt(4a^6)` refers to its non-negative square root.

Note however that `2a^3` is positive, zero or negative according to whether `a` is positive, zero or negative.

So in order to cover all real values of `a` we can write:

`sqrt(4a^6) = abs(2a^3)`

If, in addition we are told that `a >= 0` then this simplifies to:

`sqrt(4a^6) = 2a^3`