Advanced Order of Operations worksheet with eight problems to solve.

Here are the step-by-step solutions for each problem on the worksheet.

Problem 1: $-6 + (-3 - 3)^2 \div 3$
1. Parentheses: Solve inside $(-3 - 3)$ first. That equals $-6$.
* Equation becomes: $-6 + (-6)^2 \div 3$
2. Exponents: Calculate $(-6)^2$. A negative times a negative is positive, so $36$.
* Equation becomes: $-6 + 36 \div 3$
3. Division: Divide $36$ by $3$, which is $12$.
* Equation becomes: $-6 + 12$
4. Addition: Add $-6$ and $12$. This is the same as $12 - 6$.
* Result: $6$

Problem 2: $\frac{2 + 4(7 + 2^2)}{4 \times 2 + 5 \times 3}$
Treat the top (numerator) and bottom (denominator) separately.

* Top: $2 + 4(7 + 2^2)$
1. Exponent inside parenthesis: $2^2 = 4$. So, $(7 + 4)$.
2. Parenthesis: $7 + 4 = 11$.
3. Multiply: $4 \times 11 = 44$.
4. Add: $2 + 44 = 46$.
* Bottom: $4 \times 2 + 5 \times 3$
1. Multiply left side: $4 \times 2 = 8$.
2. Multiply right side: $5 \times 3 = 15$.
3. Add: $8 + 15 = 23$.
* Final Division: Divide the top by the bottom: $46 \div 23$.
* Result: $2$

Problem 3: $(5 + 9 - 10) \times 6 + 4 - 2$
1. Parentheses: Solve inside $(5 + 9 - 10)$.
* $5 + 9 = 14$.
* $14 - 10 = 4$.
* Equation becomes: $4 \times 6 + 4 - 2$
2. Multiplication: $4 \times 6 = 24$.
* Equation becomes: $24 + 4 - 2$
3. Addition/Subtraction (Left to Right):
* $24 + 4 = 28$.
* $28 - 2 = 26$.
* Result: $26$

Problem 4: $\frac{-5^2 + (-5)^2}{(4^2 - 2^5) - 2 \times 3}$

* Top: $-5^2 + (-5)^2$
* Be careful here! $-5^2$ means "negative of $5$ squared", which is $-25$.
* $(-5)^2$ means "negative $5$ times negative $5$", which is $+25$.
* Add them: $-25 + 25 = 0$.
* Bottom: $(4^2 - 2^5) - 2 \times 3$
1. Exponents: $4^2 = 16$ and $2^5 = 32$.
2. Parenthesis: $16 - 32 = -16$.
3. Multiplication: $2 \times 3 = 6$.
4. Subtract: $-16 - 6 = -22$.
* Final Division: $0 \div -22$. Zero divided by anything is zero.
* Result: $0$

Problem 5: $5 + 2^3 \times (22 \div 11) - 3^2 \times (4 + 5)$
1. Parentheses & Exponents:
* $(22 \div 11) = 2$.
* $2^3 = 8$.
* $3^2 = 9$.
* $(4 + 5) = 9$.
* Equation becomes: $5 + 8 \times 2 - 9 \times 9$
2. Multiplication:
* $8 \times 2 = 16$.
* $9 \times 9 = 81$.
* Equation becomes: $5 + 16 - 81$
3. Addition/Subtraction (Left to Right):
* $5 + 16 = 21$.
* $21 - 81 = -60$.
* Result: $-60$

Problem 6: $\{[-9 - (2 - 5)] \div (-6)\}$
1. Innermost Parentheses: $(2 - 5) = -3$.
* Equation becomes: $\{[-9 - (-3)] \div (-6)\}$
2. Brackets: $[-9 - (-3)]$ becomes $-9 + 3$, which is $-6$.
* Equation becomes: $\{-6 \div (-6)\}$
3. Division: $-6$ divided by $-6$ is positive $1$.
* Result: $1$

Problem 7: $(-7 - 5) \div [2 - 2 \cdot (-6)]$
1. First Parentheses: $(-7 - 5) = -12$.
2. Brackets: Inside $[2 - 2 \cdot (-6)]$, do multiplication first.
* $2 \cdot (-6) = -12$.
* Now subtract: $2 - (-12)$ becomes $2 + 12 = 14$.
3. Division: Divide the result from step 1 by step 2.
* $-12 \div 14$.
* Simplify the fraction: Both divide by 2. $-6/7$.
* Result: $-\frac{6}{7}$

Problem 8: $[(36 \div 6) - (-1)^3]^2 + 11$
1. Innermost Parentheses:
* $36 \div 6 = 6$.
* $(-1)^3 = -1$ (because $-1 \times -1 \times -1 = -1$).
* Equation becomes: $[6 - (-1)]^2 + 11$
2. Brackets: $6 - (-1)$ becomes $6 + 1 = 7$.
* Equation becomes: $7^2 + 11$
3. Exponents: $7^2 = 49$.
* Equation becomes: $49 + 11$
4. Addition: $49 + 11 = 60$.
* Result: $60$

──────────────────────────────────────

Final Answer:
1) 6
2) 2
3) 26
4) 0
5) -60
6) 1
7) -6/7
8) 60