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What is the value of a^4 - b^4?

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Sun Apr 28, 2019 9:58 am

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What is the value of $a^4 - b^4?$

(1) $a^2 - b^2 = 16$
(2) $a + b = 8$

[spoiler]OA=C[/spoiler]

Source: GMAT Prep

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Source: Beat The GMAT — Data Sufficiency | Sun Apr 28, 2019 9:58 am

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Sun Apr 28, 2019 10:47 am

VJesus12 wrote:What is the value of $a^4 - b^4?$

(1) $a^2 - b^2 = 16$
(2) $a + b = 8$

[spoiler]OA=C[/spoiler]

Source: GMAT Prep

Hi Vjesus12.

Let's take a look at your question.

Statement 1:

(1) $a^2 - b^2 = 16$

Using this fact we can get that $$a^4-b^4=\left(a^2-b^2\right)\left(a^2+b^2\right)=16\cdot\left(a^2+b^2\right).$$ Since we don't know the value of the second factor, this statement is NOT SUFFICIENT.

Statement 2:

(2) $a + b = 8$

Similar to the case above, we get $$a^4-b^4=\left(a^2-b^2\right)\left(a^2+b^2\right)=\left(a+b\right)\left(a-b\right)\left(a^2+b^2\right)=8\left(a-b\right)\left(a^2+b^2\right)$$ but we can't find the value of the two remaining factors. So, this statement is NOT SUFFICIENT.

Statement 1 + Statement 2:

We have that $$a^2-b^2=16\ \Rightarrow\ \left(a+b\right)\left(a-b\right)=16\ \Rightarrow\ 8\left(a-b\right)=16\ \Rightarrow\ a-b=2.$$ Hence, we get the system of equations

$a+b=8$
$a-b=2$

Adding them we get $2a=10$ which implies that $a=5$. Hence, $5+b=8$ implies that $b=3$. Therefore, $$a^4-b^4=5^4-3^2=544.$$ So, using both statements together is SUFFICIENT.

In conclusion, the correct answer is the option _C_.

I hope it is clear. <i class="em em-sunglasses"></i>

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sun Apr 28, 2019 12:44 pm

VJesus12 wrote:What is the value of $a^4 - b^4?$

(1) $a^2 - b^2 = 16$
(2) $a + b = 8$

[spoiler]OA=C[/spoiler]

Source: GMAT Prep

Target question: What is the value of aâ�´ - bâ�´?

NOTE: aâ�´ - bâ�´ is a difference of square, which we can factor. aâ�´ - bâ�´ = (aÂ² - bÂ²)(aÂ² + bÂ²)
So, we can REPHRASE the target question as...

REPHRASED target question: What is the value of (aÂ² - bÂ²)(aÂ² + bÂ²)?

Statement 1: aÂ² - bÂ² = 16
Okay, so we know the value of HALF of the target expression to get (16)(aÂ² + bÂ²), but we still don't know the value of aÂ² + bÂ², so statement 1 is NOT SUFFICIENT

We can also demonstrate that statement 1 is NOT SUFFICIENT by finding values of a and b that satisfy this condition. Here are two:
Case a: a = 4 and b = 0, in which case (aÂ² - bÂ²)(aÂ² + bÂ²) = (16)(16) = 256
Case b: a = âˆš17 and b = 1, in which case (aÂ² - bÂ²)(aÂ² + bÂ²) = (16)(18) = something other than 256
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a + b = 8
There are several values of a and b that satisfy this condition. Here are two:
Case a: a = 8 and b = 0, in which case (aÂ² - bÂ²)(aÂ² + bÂ²) = (64)(64) = 64Â²
Case b: a = 4 and b = 4, in which case (aÂ² - bÂ²)(aÂ² + bÂ²) = (0)(32) = 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 says that aÂ² - bÂ² = 16
Since we can factor aÂ² - bÂ² to get (a + b)(a - b), statement 1 is actually telling us that (a + b)(a - b) = 16
Statement 2 says that a + b = 8
So, let's take (a + b)(a - b) = 16 and replace (a+b) with 8 to get: (8)(a - b) = 16
This means that (a - b) = 2

At this point, we're done.
We now know that a - b = 2 AND we know that a + b = 8
Here we have 2 linear equations, which we COULD solve for a and b.
Once we know the exact values of a and b, we can definitely determine the value of (aÂ² - bÂ²)(aÂ² + bÂ²)
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer: C

Aside: If we solve the system, a - b = 2 and a + b = 8, we get a = 5 and b = 3. This means that (5Â² - 3Â²)(5Â² + 3Â²) = (16)(34)

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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