Root functions

A root is a number that when multiplied by itself n times equals another number. Students commonly encounter squared roots. Quite literally, a squared root is the root of a squared value. For example, the squared root of 4 is 2 or -2. The squared root of 9 is 3 or -3. The squared root of 16 is 4 or -4. In each case, the root multiplied by itself equals the original number.

Even numbered roots have a counting number root and its opposite. For example, -2^2 or 2^2 = 4, -3^2 or 3^2 = 9, and -4^2 or 4^2 = 16.

There are other root functions besides squared roots. There are cubed roots, 4th roots, 5th roots, etcetera. In each case, the value of the root is the number that when multiplied n times results in the value of the original number. Example: the cubed root of 27 = 3, because 3*3*3 = 27; the 4th root of 256 = 4 or -4, because 4*4*4*4 ( either positive or negative values) = 256, and the 5th root of 100,000 is 10, because 10*10*10*10*10 = 100,000.

Odd numbered roots of a positive value are always positive. This is because when a negative value is multiplied by itself n times, and n is odd, the result will always be negative. Example: the cubed root of 27 is 3. 3*3*3 = 27 BUT -3*-3*-3 = - 27. A negative times a negative = positive. A positive times a negative = a negative. 27 does NOT equal - 27.

Overall, roots may be positive or negative, but it is not possible to obtain a REAL number as the root of negative number. Well get into imaginary numbers like "i" later.

Roots are not always neat and tidy. They may be expressed as non-integers (1.732 is approximately the squared root of 3) or sometimes they are simply expressed as the original value under the radical symbol.

Sometimes we may be required to reduce a root to its most simplified form. To do this we must determine the factors of the original number and break it down to its constituent values. For example, the squared root of 8 is mathematically equivalent to the squared root of the values 4*2. The squared root of 4 can be reduced to 2. We place this 2 on the outside of the radical sign. The squared root of 2, however, is not an integer. We keep this under the radical sign. The simplified answer then is 2 times the squared root of 2.

Roots can be your BFF if you learn these simple rules.


Thomas D.

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