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 | = | a^{1} |
a × a | = | a^{2} (not 2 × a!) |
a× a × a | = | a^{3} (not 3 × a!) |
a× a× a× a | = | a^{4} |
a× a× a× a× a | = | a^{5} |
Some special rules apply when you divide or multiply numbers raised to some power. When you have a^{n} multiplied by a^{m}, the result is a raised to a power that is the sum of the exponents:
When you have a^{n} raised to a power m, you multiply the exponents:
Scientific calculators have a ``y^{x}'' key or a ``x^{y}'' that takes care of raising numbers to some exponent. Some fancy calculators have a ``^'' key that does the same thing. Some calculators have ``x^{2}'' and ``x^{3}'' 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!
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:
Scientific calculators have ``'' and sometimes ``'' keys to take care of the common square roots or cube roots. An expression 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:
= | a^{1/n} | |
Sqrt[a] = | = | a^{1/2} |
Cube-root[a] = | = | a^{1/3} |
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last updated: 27 May 2001