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

May 25, 2017

$\sqrt{4} {a}^{6} = 2 {a}^{3}$

#### Explanation:

$\sqrt{4 {a}^{6}} = \sqrt{4} \cdot {\sqrt{a}}^{6} = 2 {a}^{3}$

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: ${\sqrt{a}}^{6} = {a}^{\frac{1}{2}} \cdot {a}^{6} = {a}^{\frac{6}{2}} = {a}^{3}$

May 28, 2017

$2$${a}^{3}$

#### Explanation:

The expression which we have is:-

$\Rightarrow \sqrt{4 {a}^{6}}$

It can also be written as

$\Rightarrow \sqrt{4 {a}^{6}}$

$\Rightarrow \sqrt{{2}^{2} \times {a}^{6}}$

$\Rightarrow \sqrt{{2}^{2}} \times \sqrt{{a}^{6}}$

$\Rightarrow 2 \times {a}^{3}$

$\Rightarrow 2 {a}^{3}$

The final answer is: $\left(2 {a}^{3}\right)$

May 28, 2017

$\sqrt{4 {a}^{6}} = \left\mid 2 {a}^{3} \right\mid$

#### Explanation:

Note that:

${\left(2 {a}^{3}\right)}^{2} = {2}^{2} {\left({a}^{3}\right)}^{2} = 4 {a}^{6}$

So $2 {a}^{3}$ is a square root of $4 {a}^{6}$.

Note that:

${\left(- 2 {a}^{3}\right)}^{2} = 4 {a}^{6}$

So $- 2 {a}^{3}$ is a square root of $4 {a}^{6}$ too.

What do we mean when we write $\sqrt{4 {a}^{6}}$ ?

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 $4 {a}^{6} \ge 0$, so $\sqrt{4 {a}^{6}}$ refers to its non-negative square root.

Note however that $2 {a}^{3}$ 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:

$\sqrt{4 {a}^{6}} = \left\mid 2 {a}^{3} \right\mid$

If, in addition we are told that $a \ge 0$ then this simplifies to:

$\sqrt{4 {a}^{6}} = 2 {a}^{3}$