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Lesson

Roots With Even Indices

The second or square root of a given number, say x, is that number which when multiplied by itself gives x. Thus:

3 is a square root of 9 because 3 x 3 = 9 or 3² = 9
also, -3 is a square root of 9 because -3 x -3 = 9 or (-3)² = 9

Similarly

5 is a square root of 25 because 5² = 25
and, -5 is a square root of 25 because (-5)² = 25

The above examples suggest that every positive real number has two square roots, a positive one and a negative one (note that negative real numbers do not have any real square roots since no real number multiplied by itself can give a negative answer).

The positive square root of a number is called its Principal Square Root and is denoted by the symbol . Thus for example , , , and because the symbol means only the positive root.

The above examples suggest we have to be careful when dealing with variable expressions. For example what is an expression for ? We know that the symbol means the positive square root, so we cannot simply write x as the answer, because what happens if x itself is negative as in the case of (-3)² where we had ? The only way to ensure that the answer is positive is to take the absolute value of x. Then regardless of whether x is positive or negative, the answer will be positive. Thus:

The fourth root of a given number, say x, is that number which when multiplied by itself four times gives x. Thus:

2 is a fourth root of 16 because 2⁴ = 16
also, -2 is a fourth root of 16 because (-2)⁴ = 16

The above examples suggest that every positive real number has two fourth roots, a positive one and a negative one (note that negative real numbers do not have any real fourth roots since no real number multiplied by itself four times can give a negative answer).

The positive fourth root of a number is called its Principle Fourth Root and is denoted by the symbol . Thus for example , , , and .

Note the placement of the 4 in the notation for the fourth root. The 4 is called the index of the root. Note particularly its placement and size relative to the radical sign and the number under the radical sign ( which is called the radicand).

In the case of the square root the index is 2. However, the 2 is almost always omitted, thus we write instead of . It is very similar to the fact that we seldom write the exponent 1, so we usually write x and not x¹ .

As with the square root, we have to be careful when dealing with the fourth root of variable expressions. Again we cannot say that is x because of examples like . The only way to ensure that the answer is positive is to again take the absolute value of x. Then regardless of whether x is positive or negative, the answer will be positive. Thus:

This same reasoning can be extended to roots with an index of 6, 8, 10, etc. to give the following definition for the n^th root of a number when n itself is even.:

For any real number a and any even integer n greater than 1,

The above definition can also be stated in more mathematical terms as:

If a ÎÂ and nÎN then

Examples:

5. (since x² is always positive the | | signs can be removed)

Roots With Odd Indices

The third or cube root of a given number, say x, is that number which when multiplied by itself three times gives x. Thus, for example:

2 is the cube root of 8 because 2 x 2 x 2 = 8 or 2³ = 8
-2 is the cube root of -8 because -2 x -2 x -2 = -8 or (-2)³ = - 8
3 is the cube root of 27 because 3 x 3 x 3 = 27 or 3³ = 27
-5 is the cube root of -125 because -5 x -5 x -5 = -125 or (-5)³ = -125

The above examples suggest that every real number has exactly one cube root.

The mathematical symbol to denote the cube root of a number is . The 3 is called the index of the root. Note particularly its placement and size relative to the radical sign and the number under the radical sign ( which is called the radicand). For example, we can write . Since the cube root of a positive number is positive and the cube root of a negative number is negative, we can generalize and write:

The fifth root of a given number, say x, is that number which when multiplied by itself three times gives x. Thus, for example:

2 is the fifth root of 32 because 2 x 2 x 2 x 2 x 2 = 32 or 2⁵ = 32
-3 is the fifth root of -243 because -3 x -3 x -3 x -3 x -3 = - 243 or (-3)⁵ = 243

The radical notation for the fifth root is similar to that introduced for the cube root except the index is 5. Thus for example we have: . Note that the fifth root of a positive number will be positive and the fifth root of a negative number will be negative. We can thus generalize as follows:

This same reasoning can be extended to roots with an index of 7, 9, 11, etc. to give the following definition for the n^th root of a number when n itself is an odd integer,:

For any real number a and any odd integer n greater than 1,