Hi Vjesus12.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
Let's take a look at your question.
Statement 1:
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.(1) \(a^2 - b^2 = 16\)
Statement 2:
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.(2) \(a + b = 8\)
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>
