To find the value that comes in place of the question mark in the given equation, we need to follow a series of steps to evaluate each part of the expression. Let's go through them one by one.
The given equation is:
[tex]\[
\frac{\sqrt{11^3 - 31 + 12^2}}{\sqrt{6^2 \times 10 + 5^2 - 2^3 \times 3}} = ?
\][/tex]
### Step 1: Calculate [tex]\(11^3 - 31 + 12^2\)[/tex]
First, we find [tex]\(11^3\)[/tex]:
[tex]\[
11^3 = 11 \times 11 \times 11 = 1331
\][/tex]
Next, calculate [tex]\(12^2\)[/tex]:
[tex]\[
12^2 = 12 \times 12 = 144
\][/tex]
Now, substitute these values into the expression and simplify:
[tex]\[
11^3 - 31 + 12^2 = 1331 - 31 + 144 = 1331 - 31 = 1300, \quad 1300 + 144 = 1444
\][/tex]
Next, take the square root of 1444:
[tex]\[
\sqrt{1444} = 38
\][/tex]
### Step 2: Calculate [tex]\(6^2 \times 10 + 5^2 - 2^3 \times 3\)[/tex]
First, we find [tex]\(6^2\)[/tex]:
[tex]\[
6^2 = 6 \times 6 = 36
\][/tex]
Then, multiply by [tex]\(10\)[/tex]:
[tex]\[
36 \times 10 = 360
\][/tex]
Next, calculate [tex]\(5^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
Then, calculate [tex]\(2^3\)[/tex]:
[tex]\[
2^3 = 2 \times 2 \times 2 = 8
\][/tex]
Now multiply by [tex]\(3\)[/tex]:
[tex]\[
8 \times 3 = 24
\][/tex]
Substitute these values into the expression and simplify:
[tex]\[
6^2 \times 10 + 5^2 - 2^3 \times 3 = 360 + 25 - 24 = 360 + 1 = 361
\][/tex]
Next, take the square root of 361:
[tex]\[
\sqrt{361} = 19
\][/tex]
### Step 3: Divide the two results
Finally, divide the results from Step 1 and Step 2:
[tex]\[
\frac{\sqrt{11^3 - 31 + 12^2}}{\sqrt{6^2 \times 10 + 5^2 - 2^3 \times 3}} = \frac{38}{19} = 2
\][/tex]
Thus, the value that comes in the place of the question mark '?' is [tex]\( \boxed{2} \)[/tex].
