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**Exponents**

Exponents are used to denote the repeated multiplication of a number by itself; for example, 3^{4}=(3)(3) (3)(3)=81 and 3^{5}=(5) (5) (5)=125. In the expression 3^{4}, 3 is called the **base**, 4 is called the **exponent**, and we read the expression as “3 to the fourth power.” Similarly, 5 to the third power is 125.

When the exponent is 2, we call the process **squaring**. Thus, 6 squared is 36; that is, 6^{2}=(6)(6)=36. Similarly, 7 squared is 49; that is, 7^{2}=(7)(7)=49.

When negative numbers are raised to powers, the result may be positive or negative; for example, (-3)^{2}=(-3)(-3)= 9 and (-3)^{5}= (-3)(-3)(-3)(-3)(-3)= -243. A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative. Note that (-3)^{2 }= (-3)(-3)=9 but -3^{2 }= -9. Exponents can also be negative or zero; such exponents are defined as follows.

The exponent zero: For all nonzero numbers a, a^{0 }= 1. The expression 00 is undefined.

Negative exponents: For all nonzero numbers a, a^{-1 }= 1/a, a^{-2 }= 1/a^{2} , a^{-3 }= 1/a^{3} and so on. Note that (a)(a^{-1}) = (a)(1/a) = 1

**Roots**

A **square root** of a nonnegative number n is a number r such that r^{2 }= n. For example, 4 is a square root of 16 because 4^{2 }=16. Another square root of 16 is -4, since (-4)^{2}= 16. All positive numbers have two square roots, one positive and one negative. The only square root of 0 is 0. The expression consisting of the square root symbol placed over a nonnegative number denotes the nonnegative square root (or the positive square root if the number is greater than 0) of that nonnegative number. Therefore, , , . Square roots of negative numbers are not defined in the real number system.

Here are four important rules regarding operations with square roots, where a >0 and b>0.

Rule 1:

- Example A:
- Example B:

Rule 2:

- Example A:
- Example B:

Rule 3:

- Example A:
- Example B:

Rule 4:

- Example A:
- Example B:

A square root is a root of order 2. Higher order roots of a positive number n are defined similarly. For orders 3 and 4, the cube root of n, written as and fourth root of n, written as , represent numbers such that when they are raised to the powers 3 and 4, respectively, the result is n. These roots obey rules similar to those above but with the exponent 2 replaced by 3 or 4 in the first two rules.

There are some notable differences between odd order roots and even order roots (in the real number system):

- For odd order roots, there is exactly one root for every number n, even when n is negative.
- For even order roots, there are exactly two roots for every positive number n and no roots for any negative number n.

For example, 8 has exactly one cube root, but 8 has two fourth roots, and , whereas -8 has exactly one cube root, but -8 has no fourth root, since it is negative.

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