Simplify 1/sqrt(2) + 1/sqrt(2).

Step-by-step solution

1. 1

   Add the two like terms. The denominators are already identical, so the fractions add directly: $\frac{1}{\sqrt2}+\frac{1}{\sqrt2}=\frac{1+1}{\sqrt2}=\frac{2}{\sqrt2}.$ Nothing needs to be rationalised yet - and the denominator is emphatically not $2\sqrt2$, a common slip when adding fractions on autopilot.

2. 2

   Rationalise the denominator. Multiply numerator and denominator by $\sqrt2$: $\frac{2}{\sqrt2}\cdot\frac{\sqrt2}{\sqrt2}=\frac{2\sqrt2}{2}=\sqrt2.$ The $2$ in the numerator cancels the $\left(\sqrt2\right)^{2}=2$ created in the denominator.

3. 3

   See the same result a second way. Since $\dfrac{1}{\sqrt2}=\dfrac{\sqrt2}{2}$ after rationalising each term first, the sum is $\frac{\sqrt2}{2}+\frac{\sqrt2}{2}=\sqrt2.$ Rationalising before or after adding gives the same answer, which is a useful consistency check.

4. 4

   Interpret the identity $2/\sqrt2=\sqrt2$. In general $\dfrac{n}{\sqrt n}=\sqrt n$, because $n=\sqrt n\cdot\sqrt n$. So dividing a number by its own square root returns that square root - here $2\div\sqrt2=\sqrt2$.

5. 5

   Check numerically. $\dfrac{1}{\sqrt2}=0.7071067812$, so the sum is $1.4142135624$, and $\sqrt2=1.4142135624$. They agree to ten decimal places, confirming the exact form.