If you’ve been studying GRE for some time, you’ve very likely encountered the following:

.

You may, however, only seen the following equation in the context of algebra. Nevertheless, the formula above applies to number properties. Let’s take a look.

While you may be tempted to make a mad dash at it, calculating each of the squares, there is an easier way. Think of the ‘16’ as the ‘x’ and the ‘15’ as the ‘y’. Using the equation above we get:

.

Wow, that was much easier than figuring out the squares of both ‘16’ and ’15.’ Now, let’s try it for the second pair:

That leaves us with (29)(31). I know, you may be balking at my nifty little formula, thinking you still need to do some tedious multiplication. But despair not! We can still use the difference of squares formula:

is simply 3 x 3 add two zeroes: 900. Then we subtract the one and we get 899.

Next our two practice questions. The first question is not very different from the one above. The second one is more challenging and involves exponents.

## Practice Questions

1. ?

- 4
- 36
- 38
- 76
- 224

2. =

- 7
- 25
- 156
- 175
- 216

## Explanations:

** **1.

76 – 72 = 4, Answer (A).

2.

.

For the 2nd question, what happened to the numerator of the fraction? Did it just cancel out?

I understand the denominator and everything else. I just don’t know where the numerator went

Hi Sufia,

The numerator “4^8 – 3^8” gets factored into “(4^4 – 3^4) * (4^4 + 3^4)”. Then the “(4^4 + 3^4)” part of the numerator cancels out with the denominator, leaving only “(4^4 – 3^4)”. Then, the rest of the calculation you see until we arrive at the answer of “175” is essentially just the numerator as the denominator was cancelled out.

Basically, your answer of “175” is “175/1” (numerator/denominator). I hope this help! 😀

Hi,

In the 2nd question, can you explain me how did you get the(7)(25) ??

Hi Beatriz,

So, the key for this question is “the difference of squares.” As in, “x^2 – y^2 = (x – y) (x + y).”

In problem 2, you can see the steps for the difference of squares in the Explanation section. Now, once you get “(4^2 – 3^2) * (4^2 + 3^2)”, it’s a matter of calculation to get “(7) * (25)”!

= (4^2 – 3^2) * (4^2 + 3^2)

= (16 – 9) * (16 + 9)

= (7) * (25)

I am still confused by with this solution. Is not “(4^2 – 3^2)” also a difference of squares? I get:

(2 – 3)(2 +3)(4^2 + 3^2)(4^4 + 3^4) divided by (4^4 + 3^4)

(-1)(5)(16 + 9) = -125 (not even a choice)

Why do I think that “(4^2 – 3^2)” requiring further factoring?

Hi James,

You are right–(4^2 – 3^2) is also a difference of squares. If we want to break down (4^2 – 3^2), we get: (4+3)(4-3) because both of the terms are already squared. It looks like you took this one step further and took the square root of 4 as well, which is how you ended up with 2.

So let’s try this and further factor the (4^2 – 3^2). We first find the difference of squares for (4^4-3^4), which is (4^2 + 3^2)*(4^2 – 3^2)

Then, we take the difference of squares of (4^2 – 3^2), which is (4+3)*(4-3)

We multiply this by the third term from our original difference of squares:

(4+3)*(4-3)*(4^2 + 3^2)

7*1*(16+9)

7*25

So we get the same answer! Be careful when working with exponents like this–they can be tricky 🙂

How many integers between 1 and 300, inclusive, can be expressed as ‘xy’ , where x and y are integers greater than 1?

64

hi chris,

for the second question, here is what i did

(4^8/4^4) – (3^8/ 3^4) = 256 -81 = 175

is something wrong with the strategy?

HI Lawal,

You got the right answer, but technically you cannot express (a + b)/(c + d) as a/c + b/d. You would have to express it as a/(c + d) + b/(c + d).

Hope that makes sense!

A bit of easy calculation for 1st :(20+18)(20-18)-(19+17)(19-17)

=>2(38)-2(36)

=>2(38-36)=2(2)=4

Good! That’s definitely a quick and easy way to do it :).