What is the value of (64516)^(1/2)

Question

What is the value of  \sqrt{64516}

A. 256

B. 254

C. 236

D. 234

E. 266

niks18 · Accepted Answer

Hi  septwibowo 
You can use below method -

Realize that the number  \sqrt{64516}  is slightly greater than  62500 .
hence it can be broken as  = \sqrt{62500+2016} =   \sqrt{(250^{2} + 2016)}  Now this looks to me of the form  \sqrt{(a+b)^{2}} 
so can we break  2016  as  b^2 +2*a*b . this can be done as  2016 = 2*250*4 + (4)^2 
so our number becomes  = \sqrt{(250 + 4)^2}  = 254

Option  B

fitzpratik · Answer

Hi  septwibowo

One thing to notice here is that 625 is 25*25
so, 62500 will be 250*250, so our answer is greater than 250, which rules out C and D,

now numbers left are 254, 256 and 266. I start with 256 i get 256*256 = 65536, Now since 65536 > 64516, answer is 254 or B!

septwibowo · Answer

Solved it with traditional calculation

Does anyone have the more simple approach?

matthewsmith_89 · Answer

septwibowo  I would do process elimination.
Only B works out to square root of 64516.
Proof 254 x 254 = 64516
Hope this helps.

septwibowo · Answer

Thanks fitzpratik , this is exactly what I did

We can also use niks18 approach, kudos for all of you guys!

JannikHeine · Answer

Can´t you just simply recognize that 4x4 is 16 which rules out every answer choice that ends with something different than 6. Than apply the logic used above

generis · Answer

JannikHeine , I can see how it would seem logical to think that way.

The answer is: No. (6 * 6) also ends in 6. This is multiplication. We don't know what those extra digits are going to do when multiplied by each other. There are 2- 3- and x-digit numbers ending in 6 whose square ends in 16. For example 246 * 246 = 60516 196 * 196 = 38416 46 * 46 = 2116 You can use shortcuts that aren't terribly difficult. I multiplied the first two digits of each answer and compared the 3-digit result with the first three digits of the prompt, i.e., 65416 --> 654xx I knew that 25 * 25 = 625. (If not, multiply.) 625 is pretty close to 654. 25_ was still in. Like fitzpratik : C and D, with first two digits of 23, were out. If 25_ ^2 = 625xx isn't quite 654xx, 23_ ^2 would certainly not be closer. (Proof: 23 * 23 = 529, so 23_ ^2 would not even make it into the 60,000s.) Then E: (26*26) = 676. Too large. Eliminate Down to A and B. You only have to calculate one. I used B: 254 * 254 = 65416 But if I had chosen A: 256 * 256 = 65536. Too large. The right choice must be Answer B Hope it helps.

zflodeen · Answer

I think that it is also helpful to realize that it 64516 is divisible by 4 because the last 2 digits (16) are divisible by 4. To find out if it's divisible by 6, it has to be even and the sum of all the digits must be divisible by 3. It is even, but 6+4+5+1+6 = 22 which is not divisible by 3. This rules out all but B and D. Then using the 250^2 as mentioned already, you can rule out D, or you can just start multiplying the last two digits of B and D to use trial by elimination.

generis · Answer

zflodeen  -- Gah! I was so focused on unpredictable squares that I did not even think about divisibility rule for 6. That'll teach me.

That's the best shortcut, IMO. Kudos.

rahulbanerjee1992 · Answer

Here's the method I used which saved a lot time for me.

Start off by rewriting using the answer choices one by one, and removing the hundred's place digit.

A. 56

B. 54

C. 36

D. 34

E. 66

We have to pick the number which yields 16 at the end when squared.

Go for A.

56
 *56
 ---------
 XX36
 XX00
 ----------

What I did here was, I didn't calculate the entire thing because I know that the last two digits have to be 16. Here, the last two digits are 36, so eliminate the answer choice.

Go for B.

54
 *54
 ---------
 XX16
 XX00
 ----------

The last two digits here is 16, so hold onto the answer choice and repeat the same with C,D,E. The reason we repeat the method is that, C,D,E might also yield 16 at the end. In such cases, we have to do the entire calculation instead of just the last two digits like I did.

Fortunately, none of the other options yielded 16 at the end apart from Option B, so we are good. B is the answer. Hope my explanation helps!

MathRevolution · Answer

{/sqrt 64516}

=> 25 * 25 = 625. Therefore, 250 * 250 = 62500.

Suppose we have to find a square of 21. from 400 which is square of 20]

We know 20 * 20 = 400. Here, add 20 + 21 = 41 = 41 to 400 to get square of 21 as 441.

Similarly for 22: 400 + 20 + 21 + 21 + 22 = 484

Similarly, we need 64516 and we have 62500: 64516 - 62500 = 2016.

We will start from 250

250 + 251 + 251 + 252 + 252 + 253 + 253 + 254 = 8(200) + (8*50) + (16) = 1600 + 400 + 16 = 2016.

We stopped at 254.

Hence, 254 * 254 = 64516

Answer B

dortinator1234923 · Answer

What we see here is that 25*25 =625
Meaning, 250*250 will be 62500, so our answer is greater than 250, which eliminates C and D,

Now 254, 256 and 266 are left. When looking at the last two digits (tens and unit) of the square are 1 and 6, so 16. This means that the units digit of the correct answer must be 4, therefore; 254, answer B.

sarthak1701 · Answer

Same, I did the square of 256, to get an estimate of what could be the answer, it was slightly more than 64516, so 254 was the most probable but I did check 236 too just in case, and then selected 254. Took 2:40. not bad for a hard question.

Khushe · Answer

How can we rule out all options but B & D?

Since 64516 is not divisible by 6, we know that the square root will not be divisible by 6. By this logic we can only eliminate option D and no other option.

What is the value of (64516)^(1/2)

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What is the value of (64516)^(1/2)

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