I’m about to take a college entrance exam and I just can’t make sense of it all. A friend of mine called them “spicy fractions” but I genuinely do not understand. Like, what does it mean, in a physical and earthly sense when you have something like 5[Square root of 2] (which according to this, it’s 50). What is this notation for, and what does it accomplish in a real world setting? What properties apply to them, and how do you even *get* to these types of numbers?
In: Mathematics
sqrt(x) (how I’ll be writing it, same as being under the squiggly line known as a radical) is just a mathematical operation, like multiplication or addition. It determines what number, multiplied by itself (aka squared) yields this number.
So, sqrt(25) = 5, sqrt(81) = 9
However, not every number has a nice whole number answer. For example, sqrt(2) is around 1.41, but its exact value is irrational (endless decimal, no repeating). Therefore, if you want to write the exact value of sqrt(2), you just keep it written that way.
In your example, sqrt(50) is equal to 5*sqrt(2). Why? Because mathematically, we can separate terms under the same radical as multiplication.
sqrt(50)
= sqrt(25*2)
= sqrt(25)*sqrt(2)
= 5*sqrt(2)
Therefore, the exact value of sqrt(50) is 5*sqrt(2), while the approximate value is 7.07.
You can (and for test prep it would help to) try working out similar ones.
Another example:
sqrt(28)
= sqrt(7*4)
= sqrt(7)*sqrt(4)
= 2*sqrt(7)
I don’t like the term spicy fractions, but one way they’re similar is that, when you can’t get an exact value, you’re essentially simplifying it to the smallest number you can keep under the radical.
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