Category: exponents

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Someone wrote in to ask:

4⁰ * 5³ is 125. Why isn’t it 0?

On the Math.Com website, problems such as 4 to the zero power times 5 to the third power have an answer of 125 as correct. Shouldn’t the answer be zero. If not, why? Thank you!

Professor Homunculus’ response:

The answer actually should not be zero, and here’s why:

Because 4 to the 0 power is 1, not 0.

So 4⁰ * 5³ would be 1 x 5³ which is 125.

Any integer raised to the zero power equals 1.

That is hard for most people to believe, so I wrote a little piece to explain why it makes sense. Here it is:

(more…)

I am thinking about an example from a GRE (graduate record exam) book
that was shown to me.
I think it was "Which is greater, x²+y² or (x+y)²?

Here is the poop on how to think about examples like that. When in doubt – substitute
(if you can) for whole numbers. (In the original post, I had written real numbers instead of whole numbers. See the comment below about this by astute reader Randall Jones for important information about the difference that makes in this equation.)

So, try, say,  "Which is greater, 5²+3² or
(5+3)²?"
In the first case, 5² = 25 and 3² = 9, so it would be
25+9, which equals 34.
In the second case, you would first do the 5+3 (because parenthesis come first
in the order of operations) and get 8. Then you would square that, and get
64, which is clearly greater than 34.
Therefore  (5+3)²  is greater than 5²+3².

For an easy substitution you can do in your head in seconds, substitute 1s for x and for y:
= x²+y² or (x+y)²
= 1+1 or 2 ²
= 2 or 4

What if the example had been a bit different, though? What if it had been:
"Which is greater, x²*y² or (x*y)² (using multiplication instead
of addition)?

This article is continued at Mathmojo.com.