Answer :
Let's calculate each product step-by-step:
### (i) [tex]\((-4) \times (-5) \times (-8) \times (-10)\)[/tex]:
1. Multiply [tex]\(-4\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[ (-4) \times (-5) = 20 \][/tex]
2. Multiply the result by [tex]\(-8\)[/tex]:
[tex]\[ 20 \times (-8) = -160 \][/tex]
3. Finally, multiply the result by [tex]\(-10\)[/tex]:
[tex]\[ -160 \times (-10) = 1600 \][/tex]
So, the product [tex]\((-4) \times (-5) \times (-8) \times (-10) = 1600\)[/tex].
### (ii) [tex]\((-6) \times (-5) \times (-7) \times (-2) \times (-3)\)[/tex]:
1. Multiply [tex]\(-6\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[ (-6) \times (-5) = 30 \][/tex]
2. Multiply the result by [tex]\(-7\)[/tex]:
[tex]\[ 30 \times (-7) = -210 \][/tex]
3. Multiply the result by [tex]\(-2\)[/tex]:
[tex]\[ -210 \times (-2) = 420 \][/tex]
4. Finally, multiply the result by [tex]\(-3\)[/tex]:
[tex]\[ 420 \times (-3) = -1260 \][/tex]
So, the product [tex]\((-6) \times (-5) \times (-7) \times (-2) \times (-3) = -1260\)[/tex].
### (iii) [tex]\((-60) \times (-10) \times (-5) \times (-1)\)[/tex]:
1. Multiply [tex]\(-60\)[/tex] and [tex]\(-10\)[/tex]:
[tex]\[ (-60) \times (-10) = 600 \][/tex]
2. Multiply the result by [tex]\(-5\)[/tex]:
[tex]\[ 600 \times (-5) = -3000 \][/tex]
3. Finally, multiply the result by [tex]\(-1\)[/tex]:
[tex]\[ -3000 \times (-1) = 3000 \][/tex]
So, the product [tex]\((-60) \times (-10) \times (-5) \times (-1) = 3000\)[/tex].
### (iv) [tex]\((-30) \times (-20) \times (-5)\)[/tex]:
1. Multiply [tex]\(-30\)[/tex] and [tex]\(-20\)[/tex]:
[tex]\[ (-30) \times (-20) = 600 \][/tex]
2. Finally, multiply the result by [tex]\(-5\)[/tex]:
[tex]\[ 600 \times (-5) = -3000 \][/tex]
So, the product [tex]\((-30) \times (-20) \times (-5) = -3000\)[/tex].
### (v) [tex]\((-3) \times (-3) \times (-3) \times \ldots\)[/tex] 6 times:
This is equivalent to [tex]\((-3)^6\)[/tex]:
[tex]\[ (-3)^6 = 729 \][/tex]
So, the product [tex]\((-3) \times (-3) \times (-3) \times \ldots\)[/tex] 6 times is [tex]\(729\)[/tex].
### (vi) [tex]\((-5) \times (-5) \times (-5) \times \ldots\)[/tex] 5 times:
This is equivalent to [tex]\((-5)^5\)[/tex]:
[tex]\[ (-5)^5 = -3125 \][/tex]
So, the product [tex]\((-5) \times (-5) \times (-5) \times \ldots\)[/tex] 5 times is [tex]\(-3125\)[/tex].
### (vii) [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 200 times:
This is equivalent to [tex]\((-1)^{200}\)[/tex]. Since 200 is an even number:
[tex]\[ (-1)^{200} = 1 \][/tex]
So, the product [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 200 times is [tex]\(1\)[/tex].
### (viii) [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 171 times:
This is equivalent to [tex]\((-1)^{171}\)[/tex]. Since 171 is an odd number:
[tex]\[ (-1)^{171} = -1 \][/tex]
So, the product [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 171 times is [tex]\(-1\)[/tex].
Therefore, the final answers are:
[tex]\[ \begin{aligned} \text{(i)} & : 1600, \\ \text{(ii)} & : -1260, \\ \text{(iii)} & : 3000, \\ \text{(iv)} & : -3000, \\ \text{(v)} & : 729, \\ \text{(vi)} & : -3125, \\ \text{(vii)} & : 1, \\ \text{(viii)} & : -1. \end{aligned} \][/tex]
### (i) [tex]\((-4) \times (-5) \times (-8) \times (-10)\)[/tex]:
1. Multiply [tex]\(-4\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[ (-4) \times (-5) = 20 \][/tex]
2. Multiply the result by [tex]\(-8\)[/tex]:
[tex]\[ 20 \times (-8) = -160 \][/tex]
3. Finally, multiply the result by [tex]\(-10\)[/tex]:
[tex]\[ -160 \times (-10) = 1600 \][/tex]
So, the product [tex]\((-4) \times (-5) \times (-8) \times (-10) = 1600\)[/tex].
### (ii) [tex]\((-6) \times (-5) \times (-7) \times (-2) \times (-3)\)[/tex]:
1. Multiply [tex]\(-6\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[ (-6) \times (-5) = 30 \][/tex]
2. Multiply the result by [tex]\(-7\)[/tex]:
[tex]\[ 30 \times (-7) = -210 \][/tex]
3. Multiply the result by [tex]\(-2\)[/tex]:
[tex]\[ -210 \times (-2) = 420 \][/tex]
4. Finally, multiply the result by [tex]\(-3\)[/tex]:
[tex]\[ 420 \times (-3) = -1260 \][/tex]
So, the product [tex]\((-6) \times (-5) \times (-7) \times (-2) \times (-3) = -1260\)[/tex].
### (iii) [tex]\((-60) \times (-10) \times (-5) \times (-1)\)[/tex]:
1. Multiply [tex]\(-60\)[/tex] and [tex]\(-10\)[/tex]:
[tex]\[ (-60) \times (-10) = 600 \][/tex]
2. Multiply the result by [tex]\(-5\)[/tex]:
[tex]\[ 600 \times (-5) = -3000 \][/tex]
3. Finally, multiply the result by [tex]\(-1\)[/tex]:
[tex]\[ -3000 \times (-1) = 3000 \][/tex]
So, the product [tex]\((-60) \times (-10) \times (-5) \times (-1) = 3000\)[/tex].
### (iv) [tex]\((-30) \times (-20) \times (-5)\)[/tex]:
1. Multiply [tex]\(-30\)[/tex] and [tex]\(-20\)[/tex]:
[tex]\[ (-30) \times (-20) = 600 \][/tex]
2. Finally, multiply the result by [tex]\(-5\)[/tex]:
[tex]\[ 600 \times (-5) = -3000 \][/tex]
So, the product [tex]\((-30) \times (-20) \times (-5) = -3000\)[/tex].
### (v) [tex]\((-3) \times (-3) \times (-3) \times \ldots\)[/tex] 6 times:
This is equivalent to [tex]\((-3)^6\)[/tex]:
[tex]\[ (-3)^6 = 729 \][/tex]
So, the product [tex]\((-3) \times (-3) \times (-3) \times \ldots\)[/tex] 6 times is [tex]\(729\)[/tex].
### (vi) [tex]\((-5) \times (-5) \times (-5) \times \ldots\)[/tex] 5 times:
This is equivalent to [tex]\((-5)^5\)[/tex]:
[tex]\[ (-5)^5 = -3125 \][/tex]
So, the product [tex]\((-5) \times (-5) \times (-5) \times \ldots\)[/tex] 5 times is [tex]\(-3125\)[/tex].
### (vii) [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 200 times:
This is equivalent to [tex]\((-1)^{200}\)[/tex]. Since 200 is an even number:
[tex]\[ (-1)^{200} = 1 \][/tex]
So, the product [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 200 times is [tex]\(1\)[/tex].
### (viii) [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 171 times:
This is equivalent to [tex]\((-1)^{171}\)[/tex]. Since 171 is an odd number:
[tex]\[ (-1)^{171} = -1 \][/tex]
So, the product [tex]\((-1) \times (-1) \times (-1) \times \ldots\)[/tex] 171 times is [tex]\(-1\)[/tex].
Therefore, the final answers are:
[tex]\[ \begin{aligned} \text{(i)} & : 1600, \\ \text{(ii)} & : -1260, \\ \text{(iii)} & : 3000, \\ \text{(iv)} & : -3000, \\ \text{(v)} & : 729, \\ \text{(vi)} & : -3125, \\ \text{(vii)} & : 1, \\ \text{(viii)} & : -1. \end{aligned} \][/tex]
