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Your web browser needs to be able to format superscripts for the following sections! If the number ``106'' looks like ``106'', then your web browser cannot format superscripts and the following will be a bit confusing for that reason.

A shorthand way to express a quantity multiplied by itself one or more time is to use a superscript number called an exponent. So

a = a1
a × a = a2 (not 2 × a!)
a× a × a = a3 (not 3 × a!)
a× a× a× a = a4
a× a× a× a× a = a5
The quantity ``a squared'' means a2, ``a cubed'' means a3, and more generally, ``a to the nth power'' is an.

Some special rules apply when you divide or multiply numbers raised to some power. When you have an multiplied by am, the result is a raised to a power that is the sum of the exponents:

an × am = an+m
When you have an divided by am, the result is a raised to a power that is the difference of the exponents---the exponent on the bottom is subtracted from the exponent on the top:
(an) / (am) = an-m (not am-n!)

When you have an raised to a power m, you multiply the exponents:

(an)m = an × m = anm (not an+m!)
Negative exponents are used for reciprocals:
1 / a = a-1, 1 / (a2) = a-2, 1 / (a3) = a-3, 1 / (a4) = a-4, etc.

Scientific calculators have a ``yx'' key or a ``xy'' that takes care of raising numbers to some exponent. Some fancy calculators have a ``^'' key that does the same thing. Some calculators have ``x2'' and ``x3'' keys to take care of those frequent squaring or cubing of numbers. Check your calculator's manual or your instructor. The Basic Skills Computer Lab has some excellent software that can improve your skills with exponents. Try it out!


The square root of a quantity is a number that when multiplied by itself, the product is the original quantity:
Sqrt[a] × Sqrt[a] = a.
Some examples: Sqrt[1] = 1 because 1 × 1 = 1; Sqrt[4] = 2 because 2 × 2 = 4; Sqrt[38.44] = 6.2 because 6.2 × 6.2 = 38.44; Sqrt[25A2] = 5A because 5A × 5A = 25A2.

A square root of a number less than 1, gives a number larger than the number itself: Sqrt[0.01] = 0.1 because 0.1 × 0.1 = 0.01 and Sqrt[.36] = 0.6 because 0.6 × 0.6 = 0.36.

The cube root of a quantity is a number that when multiplied by itself two times, the product is the original quantity:

Cube-Root[a] × Cube-Root[a] × Cube-Root[a] = a.

Scientific calculators have ``Sqrt(x)'' and sometimes ``Cube root(x)'' keys to take care of the common square roots or cube roots. An expression Nth root(a) means the nth root of a. How can you use your calculator for something like that? You use the fact that the nth root of a is a raised to a fractional exponent of 1/n. So we have:

Nth root(a) = a1/n
Sqrt[a] = Sqrt(a) = a1/2
Cube-root[a] = Cube root(a) = a1/3
When you raise some nth root to some power m, you simply multiply the exponents as you did above for (an)m:
(Nth root(a))m = (a1/n)m = a1/n × m = am/n.
So (a6)1/2 = a6× (1/2) = a3 and (a1/2)6 = a(1/2)× 6 = a3. But if you have a1/n multiplied by am, you add the exponents since you are not raising a1/n to some power m: a1/2 × a6 = a(1/2) + 6 = a6 & 1/2 = a13/2.

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last updated: 27 May 2001

Is this page a copy of Strobel's Astronomy Notes?

Author of original content: Nick Strobel