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What's the general form for $x^y$? What's the specialized form for $x^1$ and $x^0$?

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$x^0$ is "unity" or "one". :) –  J. M. Dec 30 '10 at 6:34
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On a less frivolous note, the $x$ term (variable raised to the first power) in a polynomial is usually called the linear term. One can also speak of quintic ($x^5$), sextic ($x^6$), septic ($x^7$)... terms. –  J. M. Dec 30 '10 at 6:36
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In that context, $x^0$ (times some coefficient) would be called the constant term. –  Rahul Dec 30 '10 at 7:14
    
Also, $x^2$ is often referred to as the "quadratic" term rather than the so-called "square" term, especially in the context of polynomials. –  Alex Basson Dec 30 '10 at 12:50

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up vote 7 down vote accepted

As J.M. and Rahul Narain pointed out in comments, $x^1$ is often referred to as "linear" and $x^0$ as "constant." Commonly, though certainly not universally, $x^y$ is a/the "power" with $x$ as the base and $y$ as the exponent.

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$x$ used to be called latus (latin for side---of a square, a cube)

If you are interested in this, you must read Cajori's History of mathematical notations, which has a thorough discussion of the history of the notation and nomenclature of powers.

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