HiteshPunjabi wrote:
Which of the following is equal to 0.9998^2 ?
A. 0.99950014
B. 0.99950234
C. 0.99960004
D. 0.99960064
E. 0.99961024
M15-07
Official Solution:What is the value of \(0.9998^2\)?A. 0.99950014
B. 0.99950234
C. 0.99960004
D. 0.99960064
E. 0.99961024
APROACH 1:
Observe that the last two digits in the answer choices are different. Therefore, if we can find the last two digits of the required expression, we can identify the correct answer. The last two digits of \(0.9998*0.9998\) will be determined by the last two digits of their multiples. Next, \(98^2 = (100 - 2)^2 = 100^2 - 400 + 4 = 9604\). Therefore, the last two digits of \((0.9998)^2\) must be 04. Only option C fits.
APROACH 2:
Notice that 0.9998 is just 0.0002 away from 1: \(0.9998 + 0.0002 = 1\). How can we exploit this to simplify the calculation? If we had \(0.9998^2 - 0.0002^2\), we could apply the difference of squares formula (\(a^2 - b^2 = (a + b)(a - b)\)) and obtain \(0.9998^2 - 0.0002^2 = (0.9998 - 0.0002)(0.9998 + 0.0002) = 1*0.9996\). We don't have that, however, we could rewrite \(0.9998^2\) as \(0.9998^2 - 0.0002^2 + 0.0002^2\), which would be equal to:
\((0.9998^2 - 0.0002^2) + 0.0002^2 = \).
\(=(0.9998 - 0.0002)(0.9998 + 0.0002) + 0.0002^2 =\)
\(=1*0.9996 + 0.0002^2\) =
\(=0.9996 + 0.00000004 = 0.99960004\)
Answer: C
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