Alright, let's tackle each expression step-by-step using appropriate mathematical properties.
1. (ii) [tex]\( 21 \times 38 \times (-25) \)[/tex]:
The properties involved here are the commutative and associative properties of multiplication, indicating that the order of multiplying numbers doesn't change the result. This expression simplifies directly to:
[tex]\[
21 \times 38 \times (-25) = -19950
\][/tex]
2. (i) [tex]\( 46 \times (-38) + (-38) \times (-36) \)[/tex]:
Here, we can factor out [tex]\((-38)\)[/tex] since it is common in both terms:
[tex]\[
46 \times (-38) + (-38) \times (-36) = (-38) \times (46 - (-36))
= (-38) \times 82
= -380
\][/tex]
3. (iv) [tex]\( 25 \times (-35) \times (-4) \times 0 \)[/tex]:
Any number multiplied by zero equals zero, so this expression simplifies quickly:
[tex]\[
25 \times (-35) \times (-4) \times 0 = 0
\][/tex]
4. (iii) [tex]\( 25 \times (-6) \times (-4) \)[/tex]:
Using the associative property:
[tex]\[
25 \times (-6) \times (-4) = 25 \times 24 = 600
\][/tex]
5. (xivi) [tex]\( 625 \times (-55) + (-625) \times 65 \)[/tex]:
Here, we can factor out [tex]\( 625 \)[/tex] (with appropriate signs):
[tex]\[
625 \times (-55) + (-625) \times 65
= 625 \times (-55 - 65)
= 625 \times (-120)
= -75000
\][/tex]
6. (v) [tex]\( -36 \times 102 \)[/tex]:
This is a straightforward multiplication:
[tex]\[
-36 \times 102 = -3672
\][/tex]
7. (viii) [tex]\( (-57) \times (-19) + 57 \)[/tex]:
First, handle the multiplication, and then the addition:
[tex]\[
(-57) \times (-19) + 57 = 1083 + 57 = 1140
\][/tex]
8. (vii) [tex]\( 17 \times [100 - 2] \)[/tex]:
Simplify the expression inside the brackets first:
[tex]\[
17 \times (100 - 2) = 17 \times 98 = 1666
\][/tex]
Thus, the final results for each of the expressions are:
[tex]\[
\begin{align*}
(ii) & : -19950 \\
(i) & : -380 \\
(iv) & : 0 \\
(iii) & : 600 \\
(xivi) & : -75000 \\
(v) & : -3672 \\
(viii) & : 1140 \\
(vii) & : 1666 \\
\end{align*}
\][/tex]
