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

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

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What is the value of \(a^4 - b^4?\)

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

OA=C

Source: GMAT Prep

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VJesus12 wrote:
What is the value of \(a^4 - b^4?\)

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

OA=C

Source: GMAT Prep
Hi Vjesus12.

Let's take a look at your question.

Statement 1:
Quote:
(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:
Quote:
(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.

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VJesus12 wrote:
What is the value of \(a^4 - b^4?\)

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

OA=C

Source: GMAT Prep
Target question: What is the value of a⁴ - b⁴?

NOTE: a⁴ - b⁴ is a difference of square, which we can factor. a⁴ - b⁴ = (a² - b²)(a² + b²)
So, we can REPHRASE the target question as...

REPHRASED target question: What is the value of (a² - b²)(a² + b²)?

Statement 1: a² - b² = 16
Okay, so we know the value of HALF of the target expression to get (16)(a² + b²), but we still don't know the value of a² + b², 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 (a² - b²)(a² + b²) = (16)(16) = 256
Case b: a = √17 and b = 1, in which case (a² - b²)(a² + b²) = (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 (a² - b²)(a² + b²) = (64)(64) = 64²
Case b: a = 4 and b = 4, in which case (a² - b²)(a² + b²) = (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 a² - b² = 16
Since we can factor a² - b² 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 (a² - b²)(a² + b²)
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 (5² - 3²)(5² + 3²) = (16)(34)

Cheers,
Brent

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Brent Hanneson – Creator of GMATPrepNow.com
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