<Question>
When the two-digit integer $``\text{AB}"$ is cubed, the value is $912,\!673$. What is $A + B$?
</Question>

<think>
<Plan_1>
Okay, let's try to figure out this problem. So, we have a two-digit integer "AB", where A and B are digits, right? When this number is cubed, the result is 912,673. We need to find A + B. Hmm, okay. Let me start breaking this down.

First, AB is a two-digit number. That means A is the tens digit and B is the units digit. So, the actual numerical value is 10*A + B. For example, if A is 3 and B is 4, the number is 34. Got it. So, we need to find such a number where when you cube it, you get 912673. Then, once we find A and B, we just add them together.

The first thought that comes to my mind is to take the cube root of 912673 to see what the original number AB is. If we can find the cube root, then we can easily split it into A and B and compute their sum.

But wait, how do we find the cube root of 912673? Let me think. Since it's a six-digit number, the cube root should be a two-digit number because 100 cubed is 1,000,000, which is a seven-digit number. So, 100^3 = 1,000,000, which is bigger than 912,673. Therefore, the cube root must be less than 100. Since AB is a two-digit number, that fits because 99^3 is 970,299, which is still a six-digit number.

So, if we can find a two-digit number x such that x^3 = 912,673, then x is AB, and then A + B is the answer.

But how do we compute the cube root of 912673 without a calculator? Let's see. Maybe we can estimate it.

Let me recall some cubes.

<prompt_1.1> Estimate the cube root of 912,673 by comparing it to known cubes of two-digit numbers. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
I know that 10^3 is 1000, 20^3 is 8000, 30^3 is 27,000, 40^3 is 64,000, 50^3 is 125,000, 60^3 is 216,000, 70^3 is 343,000, 80^3 is 512,000, 90^3 is 729,000. Oh, wait, 90^3 is 729,000. But 912,673 is bigger than that. So, the cube root must be higher than 90. But since AB is a two-digit number, the maximum is 99. Wait, but 90^3 is 729k, and 912,673 is 912.673k. So, it's between 90 and 99. Hmm.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me see. Let's try 95^3. Let's compute 95^3. 95 squared is 9025, right? Then 9025 * 95. Let's do that multiplication step by step.

<prompt_2.1> Calculate 95^3 and check if it equals 857,375. </prompt_2.1>
-----

But let's verify that. Let's compute 95^3 step by step to make sure I didn't make a mistake. 95 squared: 95*95.

<prompt_2.2> Verify the calculation of 95^3 by recalculating it step by step. </prompt_2.2>
-----

But wait, let me check another way. Let's compute 95*95. 95*95: 90*90 = 8100, 90*5*2=900, 5*5=25. So, 8100 + 900 + 25 = 9025. Then 9025*95. Break it down into 9025*(100 - 5) = 9025*100 - 9025*5 = 902500 - 45125 = 857,375.

Alternatively, maybe I can check numbers around 95. For example, 94^3.

<prompt_2.3> Check if 94^3 or 96^3 equals 912,673. </prompt_2.3>

Another way: Since we know that the cube is 912,673, and we found that 95^3 is 857,375, which is less than 912,673. Then 96^3 is 884,736, which is still less. 97^3 is 912,673.

<prompt_2.4> Verify if 97^3 equals 912,673 and confirm the solution. </prompt_2.4>

But just to make sure, let's check 97*97. 97*97: 100*100 = 10,000, subtract 3*100*2 = 600, then add 3*3=9. Wait, no, that's not the right way. Let me compute 97*97 properly.

<prompt_2.5> Recalculate 97*97 to ensure accuracy. </prompt_2.5>
</Plan_2>

<Execution_2>
<execution_2.1>
First, 95 * 95 = 9025. Then, 9025 * 95. Let's break that down: 9025 * 90 + 9025 * 5. 9025 * 90: Well, 9025 * 10 is 90,250. Multiply that by 9: 90,250 * 9. Let's compute that. 90,000 * 9 = 810,000 and 250 * 9 = 2,250. So, total is 810,000 + 2,250 = 812,250. Then, 9025 * 5 is 45,125. Adding those two results together: 812,250 + 45,125 = 857,375. So, 95^3 is 857,375. But the given cube is 912,673, which is higher. So, 95 is too low.
</execution_2.1>

<execution_2.2>
Let's do 95*95. 95*95. 90*90 = 8100, 90*5*2 = 900, 5*5=25. So total is 8100 + 900 + 25 = 9025. Then 9025*95. Let's do 9025*100 = 902500, subtract 9025*5 = 45125. So 902500 - 45125 = 857,375. Wait, but the problem says the cube is 912,673. Hmm, that's not matching. So, 95^3 is 857,375, which is less than 912,673. So, maybe it's a higher number.
</execution_2.2>

<execution_2.3>
Let's see. 94^3 = 94*94*94. First, 94*94. 90*90=8100, 90*4*2=720, 4*4=16. So, 8100 + 720 + 16 = 8836. Then 8836*94. Let's compute that. 8836*90 = 795,240 and 8836*4 = 35,344. Adding them together: 795,240 + 35,344 = 830,584. That's still less than 912,673.
</execution_2.3>

<execution_2.4>
Let me check. 96^3: 96*96=9216, 9216*96. Let's compute 9216*100=921,600 minus 9216*4=36,864. So, 921,600 - 36,864 = 884,736. Then 97^3: 97*97=9409, 9409*97. Let's compute 9409*100=940,900 minus 9409*3=28,227. So, 940,900 - 28,227 = 912,673.
</execution_2.4>

<execution_2.5>
97*97: (100 - 3)*(100 - 3) = 100^2 - 2*100*3 + 3^2 = 10,000 - 600 + 9 = 9409. Then 9409*97. Let's compute that. 9409*100 = 940,900. Subtract 9409*3 = 28,227. So, 940,900 - 28,227 = 912,673.
</execution_2.5>
</Execution_2>

<Plan_3>
Based on execution_2.1:
So, maybe 98 or 99? Let's try 98.

<prompt_3.1> Calculate 98^3 and check if it equals 941,192. </prompt_3.1>
-----

Based on execution_2.4:
So, AB is 97, so A=9 and B=7. Therefore, A + B = 9 + 7 = 16.
Wait, but let me confirm once again. The cube of 97 is indeed 912,673?

<prompt_3.2> Verify the final solution and conclude the answer. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
98 squared is 9604. Then 9604 * 98. Let's compute 9604*100 = 960,400, subtract 9604*2 = 19,208. So, 960,400 - 19,208 = 941,192. That's 98^3 = 941,192. But the target is 912,673, which is still higher.
</execution_3.1>

<execution_3.2>
Yes, as we computed step by step. So, A=9, B=7, sum is 16. Therefore, the answer is 16.

But just to make sure there are no mistakes in calculation. Let's verify 97^3 again. 97*97=9409. Then 9409*97. Let's do this multiplication differently. 9409*97 = 9409*(100 - 3) = 9409*100 - 9409*3 = 940900 - 28227 = 912,673. Yes, correct. Therefore, the original number is 97, so A + B = 9 + 7 = 16.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.2:
**Final Answer**
The value of \( A + B \) is \boxed{16}.
</Plan_4>

<Execution_4>
No prompt to execute.
</Execution_4>

<Final_answer>
Therefore, $A + B = 9 + 7 = \boxed{16}$.
</Final_answer>
</think>