To solve this mathematical expression step-by-step, we should break down the expression into more manageable parts:
[tex]\[ (-7) \times 9 - (-2)^4 + \sqrt{-18 + (-2) - 9^2 \times (-3)} \][/tex]
### Step 1: Calculate [tex]\((-7) \times 9\)[/tex]
[tex]\[ (-7) \times 9 = -63 \][/tex]
### Step 2: Calculate [tex]\((-2)^4\)[/tex]
[tex]\[ (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16 \][/tex]
### Step 3: Calculate the inner terms of the square root part
We need to compute the expression inside the square root:
First, calculate the term [tex]\(9^2 \times (-3)\)[/tex]:
[tex]\[ 9^2 = 81 \][/tex]
[tex]\[ 81 \times (-3) = -243 \][/tex]
Now, consider the entire expression inside the square root:
[tex]\[ -18 + (-2) + (-243) \][/tex]
[tex]\[ -18 - 2 - 243 \][/tex]
[tex]\[ -20 - 243 = -263 \][/tex]
### Step 4: Calculate the square root part
We observe that the term inside the square root is negative:
[tex]\[ \sqrt{-263} \][/tex]
The square root of a negative number is not defined in the set of real numbers. In complex numbers, it can be expressed as:
[tex]\[ \sqrt{-263} = i\sqrt{263} \][/tex]
where [tex]\(i\)[/tex] is the imaginary unit.
### Step 5: Combine all the parts together
1. From Step 1: [tex]\(-63\)[/tex]
2. From Step 2: [tex]\(16\)[/tex]
3. From Step 4: [tex]\(i\sqrt{263}\)[/tex]
Putting all together:
[tex]\[ -63 - 16 + i\sqrt{263} \][/tex]
Simplify the real part:
[tex]\[ -63 - 16 = -79 \][/tex]
So, the final result is:
[tex]\[ \boxed{-79 + i\sqrt{263}} \][/tex]
This is the detailed solution to the given expression.
