The exponent is the little number to the upper right of a letter or numeral, like the little 2 in the expression x². It means that the number in question is to multiplied by itself that many times, thus
x² = (x)(x)
x³ = (x)(x)(x)
and so on. x² is conventionally read as "x squared." More generally, an exponent means, and can be pronounced, "x raised to the [whatever] power."Roots
One may ask the opposite question: what is the number which, when multiplied by itself [however many] times, equals x? For x² the answer is "the square root of x," symbolized by Öx ("radical x" or "root x"). For greater precision, this might be written ²Öx, but we typically do not bother; all roots are assumed to be square roots unless written otherwise. If another index, say n, is written before the radical sign, the resulting expression is pronounced "the nth root of x."Operations
Briefly, when powers of the same base are multiplied, their indices (exponents) are added, thus:
(x¹) (x²) = x³
Substitute 2 for x in these equations to demonstrate their truth.
It follows that when powers of the same base are divided, their indices are subtracted, thus
x³ /x² = x¹
It follows that:
x³ /x³ = x¹ = 1
which is to say, anything to the zero power (anything divided by itself) equals 1.
Indices of different base (as, x² times y²) cannot be multiplied; their product can only be indicated as x²y² (and so on). Powers with the same base and the same exponent (and only those) may be added, thus:
x² + x² = 2x²
When evaluating an expression like 2x², we always do the exponentiation first, and only then the multiplication. Thus, if x here equals 3, we have first
x² = 9
and then, substutiting in the expression 2x², we have
(2)(9) = 18
or in one step, 2x² = 18. The big "2" out in front is called a coefficient. Mistakes with exponents are usually mistakes of order in evaluating expressions like 2x², or mistakes in multiplication. If the above models are followed, there should never be any trouble.
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