I was tutoring my brother for calculus 2 the other day on the trapezoidal rule for area approximation, the following function:√(x2+1) when I noticed he was making a consistent horrible mistake...
He kept pulling out the +1 out of the square root to try to simplify this. So he claimed, √(1 +1)= √1+√1= 1+1 = 2.
An obvious problem with this is the claim that√2 = 2, when we know that 2×2=4 not 2.
The other problem with this is that it violates the distributive property. What do you mean Jehsuamo? These are
roots we are dealing with the distributive property is about addition and multiplication isn't it?
Yes the distributive property is a statement about addition and multiplication, but what gets lost with the root symbol may easily be seen with rational exponents.
If you already know that a(b+c) = ab +ac,
and are fine with multiplying binomials, you should know that (a+b)2 is not a2+b2
So...
read more

This journey is heavily inspired by the youtube mathematician Vi Hart, whose videos describing mathematical concepts through doodling in a notebook were the inspiration for much of my mathematical journeys series. I'll put a link to her video on this topic at the end of the journey, and I highly encourage everyone to go check her out.
Let's talk exponents.
But to do that, first we should talk about multiplication. Multiplication is a shortcut for adding a bunch of the same number together. If I gave you:
5 + 5 + 5 + 5 + 5 + 5 = ?
You could just add them normally, treating each of those 5's as a size-5 step along the number line. But since each of these addition steps is the same size, a faster way to figure out the result would be to determine two things: the size of the step, and how many steps we have. Then we can multiply the size of step (in this case, 5) by the number of steps. In this case, we have a total of 6 size-5 steps,...
read more