There are a lot of identities that are always correct. They are very useful for solving and expanding equations.
Identities about squares
First of all, there are a few identities about squares and exponentiations. For example, adding 2 squares doesn’t mean it is a square but there is an identity for that:
Think about that: Split it into 2 squares, use the formula of ab+ac=a(b+c), and finally simplify it. How about subtracting them? Well, there is another identity:
How about 2 separate squares? There is another identity. You need to know a few identities to solve most of the equations, whether you’re at school, or using this technique to solve problems in your daily life.
Also, there are some about cubes, too. Imagine you need to calculate the volume of the difference between a large and a small cube quickly. But first, we need to introduce the identity indicating the sums of cubes:
There is another identity for the difference of cubes. Remember what is said in the last paragraph? Now, that’s the answer:
Already, 5 basic identities are introduced, and you should be able to prove that algebraically although it is a bit complicated.
Examples
Imagine you facing a question in the exam: What is (y+3)(y-3)? If you don’t know the identities, you might not be able to solve that unless you suddenly proved the identity the other way round by plugging a constant for y. Also, although you might think it isn’t used when you grow up, it isn’t. Maybe someone will ask you about that at least, and use it every day at most.
Now that you know these ways to save your day, let’s memorize. For example, if someone asked what is 100-94 squared? That sounds easy because the solution is 36, which everyone knows. But if you encountered: 202-100 squared? Well, plug the identity again.
Although you might know 1022 is 10404, but how do you prove that? Well, do this if you know the square of 202 and 100. Remember the missing cube said before has an identity? Try 283 – 273 .
You will find that even if just a small margin is left, there are still a lot of cubic units. The bigger the root of exponents is, the more difference the powers have. By the way, there is also 2 identities that you must know to solve most of the equations:
You can prove this easily by logic, and you must also use this identity to prove others.