Simplify 2 divided by the second root of 2.

Step-by-step solution

1. 1

   Read the radical index. The notation $\sqrt[2]{2}$ means the second root of $2$, which is just the ordinary square root — the index $2$ is normally left off. So the expression is

   $\frac{2}{\sqrt2}$

2. 2

   Rationalise the denominator. Multiply numerator and denominator by $\sqrt2$; this is multiplying by $1$, so the value is unchanged:

   $\frac{2}{\sqrt2}\cdot\frac{\sqrt2}{\sqrt2}=\frac{2\sqrt2}{\sqrt2\cdot\sqrt2}=\frac{2\sqrt2}{2}$

   The point of the move is that $\sqrt2\cdot\sqrt2=2$ exactly, turning an irrational denominator into an integer.

3. 3

   Cancel.

   $\frac{2\sqrt2}{2}=\sqrt2$

4. 4

   Confirm in one line with exponents. Writing everything as a power of $2$:

   $\frac{2}{\sqrt2}=\frac{2^1}{2^{1/2}}=2^{1-\frac12}=2^{1/2}=\sqrt2$

   Same answer, and it explains why the result is $\sqrt2$ rather than $1$ or $2$.

5. 5

   Check numerically. $\sqrt2=1.4142136$, and $2\div 1.4142136=1.4142136$ ✓ — the number $\sqrt2$ is its own "$2$ divided by", which is just a restatement of $(\sqrt2)^2=2$.