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GRE Quant: The Difference of Squares

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

x^2 - y^2 = (x - y)(x+ y) = 0.

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.

(16^2 - 15^2)(15^2 - 14^2)

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:

(16^2 - 15^2) = (16 - 15) (16 + 15) = 31.

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

(15^2  - 14^2) = (15 - 14) (15 + 14) = 29

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:

X^2 - 1 = (x - 1)(x + 1)

30^2 - 1 = (30 - 1)(30 + 1) = (31)(29)

30^2 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. (20^2 - 18^2) - (19^2 - 17^2)?

  1. 4
  2. 36
  3. 38
  4. 76
  5. 224

2. {4^8 - 3^8}/{4^4 + 3^4}=

  1. 7
  2. 25
  3. 156
  4. 175
  5. 216

 

Explanations:

 1. (20^2 - 18^2) = (20 - 18)(20 + 18) = 76

(19^2 - 17^2) = (19 - 17)(19 + 17) = 72 

76 – 72 = 4, Answer (A).

2. {4^8 - 3^8}/{4^4 + 3^4}=

{(4^4 - 3^4)(4^4 + 3^4)}/{4^4 + 3^4}

(4^4 - 3^4) = (4^2 - 3^2)(4^2 + 3^2) = (7)(25) = 175.

 

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12 Responses to GRE Quant: The Difference of Squares

  1. Sufia February 16, 2017 at 9:47 am #

    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

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert February 16, 2017 at 12:41 pm #

      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! 😀

  2. Beatriz October 25, 2016 at 12:17 pm #

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

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert October 25, 2016 at 2:08 pm #

      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)

      • James November 6, 2016 at 9:44 am #

        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?

        • Magoosh Test Prep Expert
          Magoosh Test Prep Expert November 10, 2016 at 7:56 pm #

          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 🙂

  3. Anik May 24, 2014 at 6:25 am #

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

    • Mithun Saha May 19, 2016 at 3:16 am #

      64

  4. lawal July 24, 2012 at 10:09 am #

    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?

    • Chris Lele
      Chris July 27, 2012 at 3:16 pm #

      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!

  5. Aman July 17, 2012 at 11:39 am #

    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

    • Chris Lele
      Chris July 18, 2012 at 3:54 pm #

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


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