Let's solve the expression step by step, following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)):

Given expression:
$3\left[(-3)^2 - 2(4)^2 - (-21)\right] - \frac{6(-5)}{-3^2} - (-3)(-1)$

Step 1: Evaluate the exponents

• $(-3)^2 = 9$
• $(4)^2 = 16$
• $-3^2$ (interpreted as $-(3^2)$) = $-9$

So, the expression becomes: $3\left[9 - 2(16) - (-21)\right] - \frac{6(-5)}{-9} - (-3)(-1)$

Step 2: Multiply within the brackets and simplify

• $2(16) = 32$
• So the expression inside the brackets becomes: $9 - 32 - (-21) = 9 - 32 + 21 = -23 + 21 = -2$

So, now the expression is: $3(-2) - \frac{6(-5)}{-9} - (-3)(-1)$

Step 3: Multiply and simplify the remaining terms

• $3(-2) = -6$
• $6(-5) = -30$
• So, the next term becomes: $\frac{-30}{-9} = \frac{30}{9} = \frac{10}{3}$
• $(-3)(-1) = 3$

Now the expression is: $-6 + \frac{10}{3} - 3$

Step 4: Combine the terms

First, combine the integers: $-6 - 3 = -9$

Now, add $\frac{10}{3}$ to $-9$ (convert $-9$ to a fraction for easier addition): $-9 = \frac{-27}{3}$ $\frac{-27}{3} + \frac{10}{3} = \frac{-27 + 10}{3} = \frac{-17}{3}$

Final Answer

$\frac{-17}{3}$

The result of the expression is $\frac{-17}{3}$.

Would you like further details or explanations?

Here are five related questions to consider:

1. How does the solution change if the base of $(-3)^2$ is positive instead of negative?
2. What happens if you change the exponent of $-3$ in $-3^2$ to 3?
3. Can you evaluate $4\left[(2)^2 - 3(3)^2 - (-12)\right] - 5(-6) / -2^2 - (-2)(1)$?
4. How would the expression change if the denominator in $\frac{6(-5)}{-9}$ were positive?
5. What is the impact of changing the multiplication in $(-3)(-1)$ to division?

Tip: When dealing with expressions that include negative numbers raised to powers, be mindful of whether the negative sign is included inside or outside the parentheses, as it affects the result.