Order of Operations Question Stack Activity | Math = Love - Free Printable

To solve the given problems, we will evaluate each expression step by step. Let's go through them one by one.

---

Question 1:

\[
\frac{-5^2 - 4|2 - 4|}{2 + 3(5 - 7) + 1}
\]

#### Step 1: Simplify the numerator
- Calculate \( -5^2 \):
\[
-5^2 = -(5^2) = -25
\]
- Calculate \( |2 - 4| \):
\[
|2 - 4| = |-2| = 2
\]
- Multiply \( 4 \times |2 - 4| \):
\[
4 \times 2 = 8
\]
- Combine these results in the numerator:
\[
-5^2 - 4|2 - 4| = -25 - 8 = -33
\]

#### Step 2: Simplify the denominator
- Calculate \( 5 - 7 \):
\[
5 - 7 = -2
\]
- Multiply \( 3 \times (5 - 7) \):
\[
3 \times (-2) = -6
\]
- Add the terms in the denominator:
\[
2 + 3(5 - 7) + 1 = 2 + (-6) + 1 = 2 - 6 + 1 = -3
\]

#### Step 3: Divide the numerator by the denominator
\[
\frac{-33}{-3} = 11
\]

Answer:
\[
\boxed{11}
\]

---

Question 2:

\[
\sqrt{(-1)^5 + 17 - |9 - 5| + 4\left(3 - (1 + 2(3))\right)}
\]

#### Step 1: Simplify inside the square root
- Calculate \( (-1)^5 \):
\[
(-1)^5 = -1
\]
- Calculate \( |9 - 5| \):
\[
|9 - 5| = 4
\]
- Simplify \( 1 + 2(3) \):
\[
1 + 2(3) = 1 + 6 = 7
\]
- Simplify \( 3 - (1 + 2(3)) \):
\[
3 - 7 = -4
\]
- Multiply \( 4 \times (-4) \):
\[
4 \times (-4) = -16
\]
- Combine all terms inside the square root:
\[
(-1) + 17 - 4 + 4\left(3 - (1 + 2(3))\right) = -1 + 17 - 4 - 16 = -4
\]

#### Step 2: Take the square root
\[
\sqrt{-4}
\]
This is not a real number since the square root of a negative number is imaginary. Therefore, the expression is undefined in the real number system.

Answer:
\[
\boxed{\text{undefined}}
\]

---

Question 3:

\[
3 - 4\left(\frac{8 - (2)^3 + 4|1 - 5|}{2\sqrt{16}}\right)
\]

#### Step 1: Simplify the numerator of the fraction
- Calculate \( (2)^3 \):
\[
(2)^3 = 8
\]
- Calculate \( |1 - 5| \):
\[
|1 - 5| = |-4| = 4
\]
- Multiply \( 4 \times |1 - 5| \):
\[
4 \times 4 = 16
\]
- Combine these results in the numerator:
\[
8 - (2)^3 + 4|1 - 5| = 8 - 8 + 16 = 16
\]

#### Step 2: Simplify the denominator of the fraction
- Calculate \( \sqrt{16} \):
\[
\sqrt{16} = 4
\]
- Multiply \( 2 \times \sqrt{16} \):
\[
2 \times 4 = 8
\]

#### Step 3: Simplify the fraction
\[
\frac{16}{8} = 2
\]

#### Step 4: Multiply by 4 and subtract from 3
\[
3 - 4 \times 2 = 3 - 8 = -5
\]

Answer:
\[
\boxed{-5}
\]

---

Question 4:

\[
1 - \left|\frac{|9 - 2| - 3^2 - 1}{\sqrt{1 + 4(20)}}\right|
\]

#### Step 1: Simplify the numerator of the fraction
- Calculate \( |9 - 2| \):
\[
|9 - 2| = 7
\]
- Calculate \( 3^2 \):
\[
3^2 = 9
\]
- Combine these results in the numerator:
\[
|9 - 2| - 3^2 - 1 = 7 - 9 - 1 = -3
\]

#### Step 2: Simplify the denominator of the fraction
- Calculate \( 4 \times 20 \):
\[
4 \times 20 = 80
\]
- Add 1 to this result:
\[
1 + 80 = 81
\]
- Calculate \( \sqrt{81} \):
\[
\sqrt{81} = 9
\]

#### Step 3: Simplify the fraction
\[
\frac{|9 - 2| - 3^2 - 1}{\sqrt{1 + 4(20)}} = \frac{-3}{9} = -\frac{1}{3}
\]

#### Step 4: Take the absolute value
\[
\left| -\frac{1}{3} \right| = \frac{1}{3}
\]

#### Step 5: Subtract from 1
\[
1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\]

Answer:
\[
\boxed{\frac{2}{3}}
\]

---

Question 5:

\[
11 - (2)^2 + 4|8 - 3^2| - \frac{\sqrt{11 - 1^2 + 2(2 + 1)}}{\frac{1}{2}\left[\frac{1}{2}(8)\right]}
\]

#### Step 1: Simplify each term
- Calculate \( (2)^2 \):
\[
(2)^2 = 4
\]
- Calculate \( 3^2 \):
\[
3^2 = 9
\]
- Calculate \( |8 - 3^2| \):
\[
|8 - 9| = |-1| = 1
\]
- Multiply \( 4 \times |8 - 3^2| \):
\[
4 \times 1 = 4
\]
- Calculate \( 1^2 \):
\[
1^2 = 1
\]
- Calculate \( 2 + 1 \):
\[
2 + 1 = 3
\]
- Multiply \( 2 \times (2 + 1) \):
\[
2 \times 3 = 6
\]
- Combine these results in the square root:
\[
\sqrt{11 - 1^2 + 2(2 + 1)} = \sqrt{11 - 1 + 6} = \sqrt{16} = 4
\]
- Calculate \( \frac{1}{2}(8) \):
\[
\frac{1}{2}(8) = 4
\]
- Calculate \( \frac{1}{2} \times \frac{1}{2}(8) \):
\[
\frac{1}{2} \times 4 = 2
\]
- Divide \( \frac{\sqrt{11 - 1^2 + 2(2 + 1)}}{\frac{1}{2}\left[\frac{1}{2}(8)\right]} \):
\[
\frac{4}{2} = 2
\]

#### Step 2: Combine all terms
\[
11 - (2)^2 + 4|8 - 3^2| - \frac{\sqrt{11 - 1^2 + 2(2 + 1)}}{\frac{1}{2}\left[\frac{1}{2}(8)\right]} = 11 - 4 + 4 - 2 = 9
\]

Answer:
\[
\boxed{9}
\]

---

Question 6:

\[
8 - \left(\frac{2 - [5 - 3] + 2(6 - 1^2)}{\frac{\sqrt{18} - 2}{2}}\right)^2
\]

#### Step 1: Simplify the numerator of the fraction
- Calculate \( 5 - 3 \):
\[
5 - 3 = 2
\]
- Calculate \( 1^2 \):
\[
1^2 = 1
\]
- Calculate \( 6 - 1^2 \):
\[
6 - 1 = 5
\]
- Multiply \( 2 \times (6 - 1^2) \):
\[
2 \times 5 = 10
\]
- Combine these results in the numerator:
\[
2 - [5 - 3] + 2(6 - 1^2) = 2 - 2 + 10 = 10
\]

#### Step 2: Simplify the denominator of the fraction
- Calculate \( \sqrt{18} \):
\[
\sqrt{18} = 3\sqrt{2}
\]
- Subtract 2:
\[
3\sqrt{2} - 2
\]
- Divide by 2:
\[
\frac{3\sqrt{2} - 2}{2}
\]

#### Step 3: Simplify the fraction
\[
\frac{2 - [5 - 3] + 2(6 - 1^2)}{\frac{\sqrt{18} - 2}{2}} = \frac{10}{\frac{3\sqrt{2} - 2}{2}} = 10 \times \frac{2}{3\sqrt{2} - 2} = \frac{20}{3\sqrt{2} - 2}
\]

#### Step 4: Square the result
\[
\left(\frac{20}{3\sqrt{2} - 2}\right)^2 = \frac{400}{(3\sqrt{2} - 2)^2}
\]

#### Step 5: Simplify \( (3\sqrt{2} - 2)^2 \)
\[
(3\sqrt{2} - 2)^2 = (3\sqrt{2})^2 - 2 \cdot 3\sqrt{2} \cdot 2 + 2^2 = 18 - 12\sqrt{2} + 4 = 22 - 12\sqrt{2}
\]

#### Step 6: Combine all terms
\[
8 - \left(\frac{20}{3\sqrt{2} - 2}\right)^2 = 8 - \frac{400}{22 - 12\sqrt{2}}
\]

This expression is complex, but the exact form is:
\[
\boxed{8 - \frac{400}{22 - 12\sqrt{2}}}
\]

---

Question 7:

\[
\left|\frac{5 - 2^3}{(-1)^4 - |2^2|} - \frac{\sqrt{14 - |1 - 6|}}{-2|1 - 2| - 1^2}\right|
\]

#### Step 1: Simplify each term
- Calculate \( 5 - 2^3 \):
\[
5 - 2^3 = 5 - 8 = -3
\]
- Calculate \( (-1)^4 \):
\[
(-1)^4 = 1
\]
- Calculate \( |2^2| \):
\[
|2^2| = 4
\]
- Combine these results in the first fraction:
\[
\frac{5 - 2^3}{(-1)^4 - |2^2|} = \frac{-3}{1 - 4} = \frac{-3}{-3} = 1
\]

- Calculate \( |1 - 6| \):
\[
|1 - 6| = 5
\]
- Calculate \( \sqrt{14 - |1 - 6|} \):
\[
\sqrt{14 - 5} = \sqrt{9} = 3
\]
- Calculate \( |1 - 2| \):
\[
|1 - 2| = 1
\]
- Multiply \( -2 \times |1 - 2| \):
\[
-2 \times 1 = -2
\]
- Calculate \( -2|1 - 2| - 1^2 \):
\[
-2 - 1 = -3
\]
- Combine these results in the second fraction:
\[
\frac{\sqrt{14 - |1 - 6|}}{-2|1 - 2| - 1^2} = \frac{3}{-3} = -1
\]

#### Step 2: Combine the fractions
\[
\frac{5 - 2^3}{(-1)^4 - |2^2|} - \frac{\sqrt{14 - |1 - 6|}}{-2|1 - 2| - 1^2} = 1 - (-1) = 1 + 1 = 2
\]

#### Step 3: Take the absolute value
\[
\left| 2 \right| = 2
\]

Answer:
\[
\boxed{2}
\]

---

Question 8:

\[
4 - \sqrt{10^2 - [2(3)]^2} - 8|2 - 3(4)|
\]

#### Step 1: Simplify each term
- Calculate \( 10^2 \):
\[
10^2 = 100
\]
- Calculate \( 2(3) \):
\[
2(3) = 6
\]
- Calculate \( [2(3)]^2 \):
\[
6^2 = 36
\]
- Combine these results in the square root:
\[
\sqrt{10^2 - [2(3)]^2} = \sqrt{100 - 36} = \sqrt{64} = 8
\]

- Calculate \( 3(4) \):
\[
3(4) = 12
\]
- Calculate \( 2 - 3(4) \):
\[
2 - 12 = -10
\]
- Take the absolute value:
\[
|2 - 3(4)| = |-10| = 10
\]
- Multiply by 8:
\[
8 \times 10 = 80
\]

#### Step 2: Combine all terms
\[
4 - \sqrt{10^2 - [2(3)]^2} - 8|2 - 3(4)| = 4 - 8 - 80 = -84
\]

Answer:
\[
\boxed{-84}
\]

---

Question 9:

\[
14 - 2\left|- \sqrt{11 - 2^3 + (-1)^4}\right| - 2^3
\]

#### Step 1: Simplify inside the square root
- Calculate \( 2^3 \):
\[
2^3 = 8
\]
- Calculate \( (-1)^4 \):
\[
(-1)^4 = 1
\]
- Combine these results:
\[
11 - 2^3 + (-1)^4 = 11 - 8 + 1 = 4
\]

#### Step 2: Take the square root
\[
\sqrt{11 - 2^3 + (-1)^4} = \sqrt{4} = 2
\]

#### Step 3: Take the absolute value
\[
\left| -\sqrt{11 - 2^3 + (-1)^4} \right| = \left| -2 \right| = 2
\]

#### Step 4: Multiply by 2
\[
2 \times 2 = 4
\]

#### Step 5: Calculate \( 2^3 \)
\[
2^3 = 8
\]

#### Step 6: Combine all terms
\[
14 - 2\left|- \sqrt{11 - 2^3 + (-1)^4}\right| - 2^3 = 14 - 4 - 8 = 2
\]

Answer:
\[
\boxed{2}
\]

---

Question 10:

\[
-2 - 3(5 - 7)^3 - 4\left(\frac{16 - \sqrt{3 - 2^2}}{\sqrt{25}}\right)
\]

#### Step 1: Simplify each term
- Calculate \( 5 - 7 \):
\[
5 - 7 = -2
\]
- Calculate \( (5 - 7)^3 \):
\[
(-2)^3 = -8
\]
- Multiply by 3:
\[
3 \times (-8) = -24
\]

- Calculate \( 2^2 \):
\[
2^2 = 4
\]
- Calculate \( 3 - 2^2 \):
\[
3 - 4 = -1
\]
- Take the square root:
\[
\sqrt{3 - 2^2} = \sqrt{-1}
\]
This is not a real number since the square root of a negative number is imaginary. Therefore, the expression is undefined in the real number system.

Answer:
\[
\boxed{\text{undefined}}
\]

---

Final Answers:

\[
\boxed{11, \text{undefined}, -5, \frac{2}{3}, 9, 8 - \frac{400}{22 - 12\sqrt{2}}, 2, -84, 2, \text{undefined}}
\]