Properties of roots

The association of square roots with squares is natural. Cubic roots are associated with cubes and there isn't much complexity to it. With higher dimensions we lose the geometric visualisation and have to rely on numbers alone.

The relationship between [math]\displaystyle{ a^2 }[/math] and [math]\displaystyle{ \sqrt{a} }[/math] is the same as exponential and logarithm. One is the inverse of the other. Every time we have a number to the power of two, it means the area of a square with the length of each side side equal to that number. The square root finds the length of the side of that same square.

I think this is were two confusions happen: one is to think that the square root is halving the number. The other is to think that the square root finds the area of the square divided by 4. Both confusions have the same origin and it's roots lies in the number 2 itself. A square of side equal to 4, the area is 16. The square root of 16 is 4. The square root of 4 is 2. In a square with an area of 16 units we can fit 4 squares of area equal to 4. Now if you try the same reasoning with a side equal to 5, the square root of 5 is an irrational number and it's neither 5/2 nor 5/4. If you look at the figure, I used the variable [math]\displaystyle{ a }[/math] to represent the side. If you try multiple numbers you are going to notice that some are perfect squares while others aren't. Going further on calculus and we can study sequences of such numbers.

The previously mentioned confusion is specially common with physics. Because when we have units that are squared such as [math]\displaystyle{ m^2 }[/math] and [math]\displaystyle{ m/s^2 }[/math], a lot of interpretation mistakes originate from there.

• [math]\displaystyle{ \sqrt{a} + \sqrt{b} \neq \sqrt{a \ + \ b} \iff a^2 + b^2 \neq (a + b)^2 }[/math]. Assuming that [math]\displaystyle{ a,b \neq 0 }[/math]. The sum of roots is not the root of the sum. Conversely, the sum of squares is not the square of the sum. It's easy to see it with squares above. One very common application of this is resolution of screens, monitors and photos. If we have two monitors with the same resolution each, we didn't square the resolution, we doubled it. This is also a way to explain that doubling the length or width of a terrain does increase the total area, but much less than doubling both dimensions at the same time. This is the same informal reasoning to explain why increasing a circle's or sphere's radius has the effect of increasing the circle's perimeter or the sphere's surface much less than the circle's area or the volume of a sphere.
  
  There is another way to look at the confusion mentioned with the sum by doing this: [math]\displaystyle{ root(a) + root(b) \neq root(a + b) \iff pow(a) + pow(b) \neq pow(a + b) }[/math]. I wrote the same thing but replaced the symbols with functions. Some functions may obey to that property but that is not something we study in calculus. Functional analysis studies such properties more thoroughly.

• [math]\displaystyle{ \sqrt{a}\sqrt{b} = \sqrt{ab} }[/math]. Remember that roots are rational exponents. Then [math]\displaystyle{ a^{\frac{1}{2}}b^{\frac{1}{2}} = (ab)^{\frac{1}{2}} \iff (a^{\frac{1}{2}}b^{\frac{1}{2}})^2 = ((ab)^{\frac{1}{2}})^2 \iff a^1 b^1 = (ab)^1 }[/math] which proves the property.
  
  
  
  
  I'm going to repeat the same squares from above with a slight change. Now the root is the side and the variable is the area itself. Most people are familiar with this property [math]\displaystyle{ 2\sqrt{x} = \sqrt{2^2 x} }[/math]. In other words, we move a number under the nth root by rewriting the number with the nth power. In the previous figure we looked at the areas, let's take a look at perimeters now: the perimeter of the large square is [math]\displaystyle{ 4\sqrt{a} }[/math]. Now let's see the perimeter of one tile: [math]\displaystyle{ 2\sqrt{a} = \frac{\sqrt{a}}{2} + \frac{\sqrt{a}}{2} + \frac{\sqrt{a}}{2} + \frac{\sqrt{a}}{2} = \sqrt{4a} = \sqrt{4}\sqrt{a} }[/math]. See? That property of putting a number from outside the root to under it is related to the perimeters, not the areas.
  
  Reference: Vatsal Ojha. I was searching for a graphical interpretation for the property above and found one and the mentioned name. The idea came from him.