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Explain the following properties:
i) ($-$a1) $\times$ ($-$a2) $\times$ ($-$a3) $\times$ ... $\times$ ($-$an) = $-$ (a1 $\times$ a2 $\times$ a3 $\times$ ... $\times$ an), when n is odd.ii) ($-$a1) $\times$ ($-$a2) $\times$ ($-$a3) $\times$ ... $\times$ ($-$an) = (a1 $\times$ a2 $\times$ a3 $\times$ ... $\times$ an), when n is even.iii) ($-$a) $\times$ ($-$a) $\times$ ($-$a) $\times$ ... n times = $-$ an, when n is odd. iv) (-$a) $\times$ ($-$a) $\times$ ($-$a) $\times$ ... n times = an, when n is even.
v) ($-$1) $\times$ ($-$1) $\times$ ($-$1) $\times$ ... n times = $-$ 1, when n is odd.v) ($-$1) $\times$ ($-$1) $\times$ ($-$1) $\times$ ... n times = 1, when n is even.
Given:
Some properties.
To do:
We have to explain the properties given.
Solution:
i) ($-$a1) $\times$ ($-$a2) $\times$ ($-$a3) $\times$ ... $\times$ ($-$an) = $-$ (a1 $\times$ a2 $\times$ a3 $\times$ ... $\times$ an), when n is odd.
Explanation: If odd number of negative numbers are multiplied then the product will always be a negative number.
Example: ($-$1) $\times$ ($-$2) $\times$ ($-$3) = $-$ 6
ii) ($-$a1) $\times$ ($-$a2) $\times$ ($-$a3) $\times$ ... $\times$ ($-$an) =
(a1 $\times$ a2 $\times$ a3 $\times$ ... $\times$ an), when n is even.
Explanation: If even number of negative numbers are multiplied, then the product will always be a positive number.
Example: ($-$1) $\times$ ($-$2) $\times$ ($-$3) $\times$ ($-$4) = 24
iii) ($-$a) $\times$ ($-$a) $\times$ ($-$a) $\times$ ... n times = $-$ an, when n is odd.
Explanation: If a negative number is multiplied by itself an odd number of times, then the product will be negative of that number raised to the power number of times it was multiplied.
Example: ($-$2) $\times$ ($-$2) $\times$ ($-$2) = $-$ 23
iv) ($-$a) $\times$ ($-$a) $\times$ ($-$a) $\times$ ... n times = an, when n is even.
Explanation: If a negative number is multiplied by itself an even number of times, then the product will be positive of that number raised to the power number of times it was multiplied.
Example: ($-$2) $\times$ ($-$2) $\times$ ($-$2) $\times$ ($-$2) = 24
v) ($-$1) $\times$ ($-$1) $\times$ ($-$1) $\times$ ... n times = $-$ 1, when n is odd.
Explanation: If ($-$1) is multiplied by itself an odd number of times, then the product will be ($-$1).
Example: ($-$1) $\times$ ($-$1) $\times$ ($-$1) = $-$ 1
vi) ($-$1) $\times$ ($-$1) $\times$ ($-$1) $\times$ ... n times = 1, when n is even.
Explanation: If ($-$1) is multiplied by itself an even number of times, then the product will be (1).
Example: ($-$1) $\times$ ($-$1) $\times$ ($-$1) $\times$ ($-$1) = 1
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