Solve the Nested Square Root: Finding √√12

Solve the following exercise:

$\sqrt{\sqrt{12}}=$

In order to solve the following expression $\sqrt{\sqrt{12}}$, it needs to be simplified using the properties of exponents and roots. Specifically, we apply the rule that states that the square root of a square root can be expressed as a fourth root.

Let's break down this solution step by step:

• First, represent the inner $\sqrt{12}$ as a power: $12^{1/2}$.

• Next, take the square root of this result, which involves raising $12^{1/2}$ to the power of $1/2$ again:
  $\left(12^{1/2}\right)^{1/2} = 12^{(1/2) \cdot (1/2)} = 12^{1/4}$.

• According to the rules of exponents, raising an exponent to another power results in multiplying the exponents.

• This gives us $12^{1/4}$, which we can write as the fourth root of 12: $\sqrt[4]{12}$.

In conclusion the simplification of $\sqrt{\sqrt{12}}$ is $\sqrt[4]{12}$.