Question

create a rule or identity to describe those patterns. For example, if you notice that every time you multiply a negative number by another negative number the result is positive, we can generalize this by saying (-a)(-b) = c, where a, b, and c are all positive real numbers.

Answer 1

This equation generalizes the patterns seen in part D, where a and b represent real numbers:

(a + bi)(a − bi) = a2 + b2.

Answer 2

a/b = a * 1/b

Explanation:

a/b = a * 1/b

this looks easy....

But what about

a / (1/b) = a*b

if you see any division, you can always multiply BOTH the numerator AND the denominator by the same amount. That is allowed because that is the same as multiiplying by the number 1.

So multiply by b/b and see what happens.

a / (1/b) = b/b * a / (1/b)

group the numerator AND the denominator

a / (1/b) = (b * a ) / b*(1/b)

a / (1/b) = (b * a ) / b/b

a / (1/b) = (b * a ) / 1

a / (1/b) = (b * a )

a / (1/b) = a*b

It is allowed because you can always multiply any number by 1 and that is because that will not change the original number. The number multiplied by 1 remains the same number.

This notion is very powerful and it can be used to turn any division into a multiplycation, and vice versa.

a / (1/b) = a*b