Solve these eight arithmetic challenges by applying the distributive property to simplify the expressions.

The worksheet focuses on applying the distributive property to simplify and solve mathematical expressions. The distributive property states:

\[
a \times (b + c) = a \times b + a \times c
\]

We will solve each problem step by step using this property.

---

Problem 1: \( 44 \times 8 + 44 \times 92 \)

Using the distributive property:
\[
44 \times 8 + 44 \times 92 = 44 \times (8 + 92)
\]

Simplify inside the parentheses:
\[
8 + 92 = 100
\]

So:
\[
44 \times (8 + 92) = 44 \times 100 = 4400
\]

Answer:
\[
\boxed{4400}
\]

---

Problem 2: \( 4 \times 3 \times 12 \times 966 + 144 \times 34 \)

First, simplify the left part of the expression:
\[
4 \times 3 \times 12 \times 966 = 144 \times 966
\]

So the expression becomes:
\[
144 \times 966 + 144 \times 34
\]

Now apply the distributive property:
\[
144 \times 966 + 144 \times 34 = 144 \times (966 + 34)
\]

Simplify inside the parentheses:
\[
966 + 34 = 1000
\]

So:
\[
144 \times (966 + 34) = 144 \times 1000 = 144000
\]

Answer:
\[
\boxed{144000}
\]

---

Problem 3: \( 127834 \times 1001 - 127834 \)

Factor out \( 127834 \):
\[
127834 \times 1001 - 127834 = 127834 \times (1001 - 1)
\]

Simplify inside the parentheses:
\[
1001 - 1 = 1000
\]

So:
\[
127834 \times (1001 - 1) = 127834 \times 1000 = 127834000
\]

Answer:
\[
\boxed{127834000}
\]

---

Problem 4: \( 9999 \times 7 + 3 \times 9999 \)

Factor out \( 9999 \):
\[
9999 \times 7 + 3 \times 9999 = 9999 \times (7 + 3)
\]

Simplify inside the parentheses:
\[
7 + 3 = 10
\]

So:
\[
9999 \times (7 + 3) = 9999 \times 10 = 99990
\]

Answer:
\[
\boxed{99990}
\]

---

Problem 5: \( 16 \times 16 + 2 \times 8 \times 4 \)

First, simplify the right part of the expression:
\[
2 \times 8 \times 4 = 64
\]

So the expression becomes:
\[
16 \times 16 + 64
\]

Calculate \( 16 \times 16 \):
\[
16 \times 16 = 256
\]

Add the results:
\[
256 + 64 = 320
\]

Answer:
\[
\boxed{320}
\]

---

Problem 6: \( 3 \times 15 \times 6 + 4 \times 45 \)

First, simplify the left part of the expression:
\[
3 \times 15 \times 6 = 270
\]

So the expression becomes:
\[
270 + 4 \times 45
\]

Simplify the right part:
\[
4 \times 45 = 180
\]

Add the results:
\[
270 + 180 = 450
\]

Answer:
\[
\boxed{450}
\]

---

Problem 7: \( 49 \times 1444 - 444 \times 7 \times 7 \)

First, simplify the right part of the expression:
\[
7 \times 7 = 49
\]
\[
444 \times 49 = 21756
\]

So the expression becomes:
\[
49 \times 1444 - 21756
\]

Simplify the left part:
\[
49 \times 1444 = 70756
\]

Subtract the results:
\[
70756 - 21756 = 49000
\]

Answer:
\[
\boxed{49000}
\]

---

Problem 8: \( 2 \times 78361 - 836198 \times 0 \)

Simplify the right part of the expression:
\[
836198 \times 0 = 0
\]

So the expression becomes:
\[
2 \times 78361 - 0 = 2 \times 78361
\]

Calculate:
\[
2 \times 78361 = 156722
\]

Answer:
\[
\boxed{156722}
\]

---

Final Answers:

\[
\boxed{4400, 144000, 127834000, 99990, 320, 450, 49000, 156722}
\]