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
Alright so you know how 5^2 = 25 ? This is called squaring a number: multiplying it by itself. The square root is the opposite of this: square root 25 = 5.
Now, the square root of 50.
Square root(50) = square root(25 x 2)
square root(25 x 2) = square root(25) x square root(2)
square root(25) x square root(2) = 5 x square root(2).
Happy to help if you have further questions!
Your example is incorrect, we might be able to help better if you find the actual problem your having troubles with.
Edit: do you mean 5 SQUARED, times 2?
How you do it:
You’re supposed to factor the number. For example, 6 = 2*3 and 15 = 3*5. When you’re asked to take the square root of 50, but you realize there’s no whole number answer, instead you can factor 50 into 5*5*2. Then you can break sqrt(50) into sqrt(5*5) * sqrt(2), or 5 * sqrt(2).
What is this notation for?
It’s the way to write an answer in its simplest form.
As a real-world problem, let’s say you have a right triangle and the two smaller sides are length 5. What is the length of the hypotenuse?
The pythagorean theorem says that a^2 + b^2 = c^2. Since the two sides are 5 and 5, square those and you get 25 and 25, so the hypotenuse squared is 50. So to get the hypotenuse you need the square root of 50.
However, just like when we write fractions we think it’s better to write 2/3 instead of 4/6, when writing answers involving square roots we think it’s better to write 5 sqrt(2) instead of sqrt(50).
its lets you simplify square roots without giving up exact answers. Say you have √50/√98, what even is that in normal numbers? No idea, so simplify both. you end up with 5√2 / 7√2, the √2s cancel, leaving you with just 5/7.
To get them you just do division with perfect squares. √50=√(25*2)=√25*√2=5√2
This works for pretty much any number, (it must be the product of at least 1 square though)
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.
As a physical and earthly thing: draw a line. If you make that line into a square, the area of the square is the length of the line squared. I.e., a square with sides that are 5 feet long has an area of 5*5=25 square feet.
Finding the square root is doing it in reverse. If you have a square that has an area of 50 square feet, then how long is one of the sides? Answer: The square root of 50. (I.e. 5 times the square root of 2).
Then you start finding squares and square roots in all sorts of geometry, which gets more complicated but is built in this foundation.
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