The decimal representation of 12.025 has 5 digits. If x = 18,000,000

Question

The decimal representation of  12.025  has 5 digits. If  x = 18,\!000,\!000,\!000.5 , how many digits are in the decimal representation of  x^4 ?

(A) 42
(B) 43
(C) 44
(D) 45
(E) 46

kevincan · Answer

If you get this question wrong , let us know your line of reasoning

JamesGMAT2026 · Answer

Was really unsure about an efficient method on this one but I tried as follows

x = 18 * 10^9 + 0.5 
let a = 18*10^9 , b = 0.5 
x^2 = (a+b)^2 = a^2 + b^2 + 2ab = 3.24 * 10^20 + 1.8*10^10 + 0.25

then x^4 = (3.24 * 10^20 + 1.8*10^10 + 0.25)^2

to find the number of digits in this as quickly as possible only consider (3.24*10^20)^2 and 0.25^2 as the digits of the rest will fit in between 
 3.24^2 = 10.4976  
 10^20^2 = 10^40

(3.24*10^20 )^2 = 1.04976*10^41
0.25^2 = 0.0625 
Hence we have 41 + 5 decimal places = 46 digits

If anyone has a quicker method this would be quite helpful

_____
James (NL)

KindFleet · Answer

Hey James,

Can you please let me know how 0.0625 is being considered 5 digits instead of 4 coz unit digit is already included in that 41.

gouravkr · Answer

We know 0.5 ^ 4 will give 4 digits, so put that aside.
We know the 9 zeroes raised to the power of 4 will give 36 digits, so put that aside.

The question now boils down to how many digits are there in 18^4. I see no way to but to just calculate it.
First do 18x18 = 324, then 324 x 324 = 104976. This is 6 digits.
So 6 + 36 + 4 = 46 digits.

This is one way to solve that I could think of.

Edit: Square of a number with n digits can either have 2n - 1 or 2n digits. If you're familiar with the cutoff points (33 for 2 digit numbers, 316 for 3 digit numbers, and 3162 for 4 digit numbers), this problem becomes much easier.
since 324 > 316, so 324^2 has 6 digits and you don't need to calculate the square of 324.

gouravkr · Answer

The explanation here is incorrect. 1 x 10^1 gives 2 digits, 1 x 10^2 gives 3 digits, so 1 x 10^41 gives 42 digits. There are only 4 decimal digits. So 42 + 4 = 46

egmat · Answer

Hi KindFleet,

You're actually reading 0.0625 correctly. Its fractional part is  4  digits: 0, 6, 2, 5. The trouble is just in how James split the total, so let me line it up cleanly.

Where the labels slipped

James wrote it as  41  +  5  =  46 , but the cleaner split is  42  +  4  =  46 :

-  Integer part:  1.04976 × 10^41. Since 10^41 is a 1 followed by  41  zeros, that's a  42 -digit whole number - not  41 . (This is the off-by-one that made his count look strange.)
-  Fractional part:  .0625 - the digits 0, 6, 2, 5 =  4  digits.

So the real number looks like  .0625, and total digits =  42  +  4  =  46 .

About that leading "0"

Here's the key point behind your question. When we write the fractional part as 0.0625, the  0 before the decimal is not a digit of the number  - it's just a placeholder standing in for "the integer part." The actual integer part isn't 0; it's that huge  42 -digit number. So there's no double-counting: the units digit lives inside the  42 , and the four digits 0625 live to the right of the decimal. They're completely separate.

Your instinct that 0.0625 is  4  digits is right - the fix is simply that the integer side is  42 , not  41 .

Quick check to lock it in

Take a small stand-in:  500.0625 .
- Integer part 500 -  3  digits.
- Fractional part 0625 -  4  digits.
- Total =  7  digits, and the "0" in 0.0625 never appears because 500 replaced it.

Same structure as your problem, just smaller.

Answer: E

The decimal representation of 12.025 has 5 digits. If x = 18,000,000

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The decimal representation of 12.025 has 5 digits. If x = 18,000,000

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