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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Roots and Powers

## The Relationship Between Roots and Powers

Letâ€™s look again at the relationship between taking a square root and squaring.

 For example, letâ€™s start with the number 6. 6 Take its square root. Square the square root. Write the product as a single radical. Multiply. Simplify. = 6 The result, 6, is the number that we started with.
Thus, squaring â€œundoesâ€ taking a square root.

Here, we saw that = 6.

This is true in general.

Property â€” Raising an nth Root to the nth Power

English Raising the nth root of a number to the nth power results in the original number.

Algebra If is defined, then Here, n is a positive integer.

Example

Example 1

Simplify:

Solution

Use this relationship to simplify each radical:

Note:

only when x 0. An even root of a negative number is not a real number.

Now, letâ€™s see what happens when we first square a negative number, and then take its square root.
 For example, letâ€™s start with the number -4. Square the number. The result is positive. -4(-4)2 = 16 Take the square root. Simplify. = 4

The result, 4, is the opposite of the number we started with.

Thus, when we square a negative number, and then take the square root, we obtain the opposite of the original number.

This relationship holds true, in general for even roots.