Algebra/Roots and Radicals

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Contents 1 Square root 2 Cube roots 3 Other roots 4 Irrational numbers

[edit] Square root

If you take the square root of a number the result is the number which when squared gives the first number. This can be written symbolically as:

. Since no matter what the value of y is, we cannot define the when x < 0.

Examples:

• since
• If x = 5 then .
• If x = 7 then is undefined because , but there is no number y so that y² = − 3 (recall that a negative number times a negative number is a positive number). Notice that the answer is not - 3. Why not? The answer is simply that and not − 9.

[edit] Cube roots

Roots do not have to be square. One can also take the cube root of a number ( ) of a number is the number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. For example, the cube root of 8 is 2, because , or:

[edit] Other roots

There are an infinite number of possible roots all in the form of which corresponds to , when expressed using indices. If then .

[edit] Irrational numbers

If you square root a whole number which is not itself the square of a rational number the answer will have an infinite number of decimal places. Such a number is described as irrational and is defined as a number which cannot be written as a rational number: , where a and b are integers.

However, using a calculator you can approximate the square root of a non-square number:

The result of taking the square root is written with the approximately equal sign because the result is an irrational value which cannot be written in decimal notation exactly. Writing the square root of 3 or any other non-square number as is the simplest way to represent the exact value.