Which of the following is equal to 0.9998^2 ?

I began to multiply .9998 x .9998. I knew the units digit would be 4 (8 x 8 = 64) and then I figured out the hundreds digit by multiplying 9 x 8 + 6 (from the 64) = 84. Rather than doing the rest of the math, I skipped to the tens digit in the second figure. That gave me 9 x 8 = 72 or 2 in the tens column plus the 8 gave me 04 for the first two digits. Figured out it was C.

\((0.9998)^2=(1-0.0002)^2=(1-2*10^-^4)^2=1+2^2*10^-^8-2*2*10^-^4\) it's the old formula \((a+b)^2=a^2+b^2+2ab\)

\(1+4*10^-^8-4*10^-^4=1+0.00000004-0.0004\)

\((1-0.0004)+0.00000004\)
\(0.9996+0.00000004=0.99960004\)
C

thnks Zarrolu
i followed the same method....bt i missed ...thnks maan

Archit

It just faster to do criss cross multiplication in this question and you can get the answer in less than 30 secs.

9 9 9 8

9 9 9 8

Units place : 8 x 8 = 64 => Units place is 4 with carry over of 4

Tens place : 9x8 + 9x8 + 6 = 150 => Tens place is 0 with carry over of 15

You can stop here since you have already found answer by checking options

This is criss cross multiplication technique from vedic mathematics (https://www.brighthubeducation.com/homew ... thematics/) and i would highly recommend looking it up.

Elaborating 100's place : 9x8 + 9x9 + 9x8 +15 = 240 => 100s place is 0 with carry over 24

1000 place : 9x8 +9x9 + 9x9 + 9x8 + 24 = 330 => 1000 place is 0 with carry over 33 and so on....

For the sake of convenience, let us assume it to be= 9998*9998 = 9998*(10000-2) = 99980000-19996 --> the units digit is 4 and the tens a 0. Only C satisfies.

C.

Multiplying

0.9998
0.9998
______
------84
----82
______
------04 in the end


only C

Just for sake of convenience, remove decimal from the question as well from the answer

\(9998^2 = (10^4 - 2)^2 = 100000000 + 4 - 40000 = 100000004 - 40000\)

Last 2 digits are 04

Answer = C

Hi,

You just need to consider the 2 last digits to find the right answer.
98^2=9604 =>look at the 2 last digits it's 04, therefore only Answer C) satisfies.

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

We can use the identity:

(a - b)^2 = a^2 - 2ab + b^2

In this case, we can let a = 1 and b = 0.0002 (since 0.9998 is 1 - 0.0002):

(1 - 0.0002)^2 = 1^2 - 2(1)(0.0002) + 0.0002^2
= 1 - 0.0004 + 0.00000004
= 0.99960004

Therefore, 0.9998^2 is equal to 0.99960004. Option C is correct.