Advanced Order of Operations worksheet featuring eight problems requiring the application of PEMDAS rules to solve expressions with integers, exponents, and parentheses.

Let's solve each problem step by step using the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

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Problem 1:

$$
-6 + (-3 - 3)^2 \div 3
$$

Step 1: Simplify inside parentheses:
$$
-3 - 3 = -6
$$

Step 2: Apply exponent:
$$
(-6)^2 = 36
$$

Step 3: Divide:
$$
36 \div 3 = 12
$$

Step 4: Add:
$$
-6 + 12 = 6
$$

✔ Answer: 6

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Problem 2:

$$
\frac{2 + 4(7 + 2^2)}{4 \times 2 + 5 \times 3}
$$

Numerator:
- First, exponent: $2^2 = 4$
- Then: $7 + 4 = 11$
- Multiply: $4 \times 11 = 44$
- Add: $2 + 44 = 46$

Denominator:
- $4 \times 2 = 8$
- $5 \times 3 = 15$
- $8 + 15 = 23$

Now divide:
$$
\frac{46}{23} = 2
$$

✔ Answer: 2

---

Problem 3:

$$
(5 + 9 - 10) \times 6 + 4 - 2
$$

Step 1: Simplify inside parentheses:
$5 + 9 = 14$, $14 - 10 = 4$

Step 2: Multiply:
$4 \times 6 = 24$

Step 3: Add and subtract left to right:
$24 + 4 = 28$, $28 - 2 = 26$

✔ Answer: 26

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Problem 4:

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

⚠️ Important: Note the difference between $-5^2$ and $(-5)^2$

- $-5^2 = -(5^2) = -25$ (exponent first, then negative)
- $(-5)^2 = 25$

So numerator:
$$
-25 + 25 = 0
$$

Denominator:
- $4^2 = 16$
- $2^5 = 32$
- $16 - 32 = -16$
- $2 \times 3 = 6$
- $-16 - 6 = -22$

Now:
$$
\frac{0}{-22} = 0
$$

✔ Answer: 0

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Problem 5:

$$
5 + 2^3 \times (22 \div 11) - 3^2 \times (4 + 5)
$$

Step 1: Exponents:
- $2^3 = 8$
- $3^2 = 9$

Step 2: Parentheses:
- $22 \div 11 = 2$
- $4 + 5 = 9$

Now substitute:
$$
5 + 8 \times 2 - 9 \times 9
$$

Step 3: Multiplication:
- $8 \times 2 = 16$
- $9 \times 9 = 81$

Now:
$$
5 + 16 - 81 = 21 - 81 = -60
$$

✔ Answer: -60

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Problem 6:

$$
\left\{[-9 - (2 - 5)] \div (-6)\right\}
$$

Step 1: Inside innermost parentheses:
$2 - 5 = -3$

Now:
$-9 - (-3) = -9 + 3 = -6$

Now divide:
$-6 \div (-6) = 1$

✔ Answer: 1

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Problem 7:

$$
(-7 - 5) \div [2 - 2 - (-6)]
$$

Numerator:
$-7 - 5 = -12$

Denominator:
$2 - 2 = 0$, $0 - (-6) = 0 + 6 = 6$

Now:
$$
-12 \div 6 = -2
$$

✔ Answer: -2

---

Problem 8:

$$
[(36 \div 6) - (-1)^3]^2 + 11
$$

Step 1: $36 \div 6 = 6$

Step 2: $(-1)^3 = -1$ (odd power)

Now:
$6 - (-1) = 6 + 1 = 7$

Step 3: Square it:
$7^2 = 49$

Step 4: Add 11:
$49 + 11 = 60$

✔ Answer: 60

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✔ Final Answers:

1. 6
2. 2
3. 26
4. 0
5. -60
6. 1
7. -2
8. 60

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