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Let's look at a number...  8...

8

What number can we multiply to 8 to get 1 (the multiplicative identity) as the answer?

8 x __ = 1

Wow!  That's a really weird problem.  Can you think of any integers that would work?  Nope.  So...  It must be a fraction!  Remember that we want 1 for the answer... and 1 in fraction language with 8's is

8 / 8 = 1
 

8 x 1/8 = 8/1 x 1/8 = 8/8 = 1

8 x 1/8 = 1

So, the multiplicative inverse of 8 is 1/8!

Let's go the other way...

What number can we multiply to 1/8 to get 1 as the answer?

1/8 x __ = 1

Remember this!

8/8 = 1
 

1/8 x 8 = 1

So, the multiplicative inverse of 1/8 is 8!