The value of (2^0+3^-1) × 3^22 is:
Answers
Step-by-step explanation:
we know that 2×0=0. Intuitively this is because if you add 2 to itself zero times, you get zero. Or, to be concrete, if someone gives you two apples zero times, you have zero apples.
For repeatedly adding 2 ,
talking about collections of apples is a good model. But for repeatedly multiplying by 2, it isn't necessarily, since you can't multiply apples and apples. But you can multiply apples by numbers; that is, you can start with 1 apple, then double the number of apples you have to get 2 apples, then double the number of apples you have to get 4 apples, and so forth. In general if you double your apples n times, you have 2n apples.
What happens if you double your apples zero times?
Well, that means you haven't started doubling them yet, so you still have 1
apple. If you want your notation to be consistent, then you should say 2^0=1
It's intuitive what it means to add different amounts of apples, and it's intuitive what it means to have zero apples. But the twos I am now working with aren't numbers of apples, they're just abstract numbers; in other words, they're unitless, so it's harder to get a grip on them.
What 2n really represents above is an endomorphism of the free commutative monoid on an apple, which is much less concrete than an apple.
There is a way to gain intuition here which sort of involves units. One way to interpret 2n is that it is the "size" of an n-cube of side length 2 in dimension n. For example, the length of a segment of side length 2 is 2, the area of a square of side length 2 is 4, and so forth. One way to say this is that 2n is the number of n-cubes of side length 1 that fit into an n-cube of side length 2.
To get a meaningful interpretation of the above when n=0
we need to decide what 0-dimensional objects are. Well, if 2-dimensional space is a plane and 1-dimensional space is a line, then 0-dimensional space must be... a point! In particular, a 0-cube, of any side length, is a point, and so exactly one 0-cube of side length 1 fits into a 0-cube of side length 2.
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