1.3. Exponents and roots

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Exponents are used to denote the repeated multiplication of a number by itself; for example, 34=(3)(3) (3)(3)=81 and 35=(5) (5) (5)=125. In the expression 34, 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, 62=(6)(6)=36.  Similarly, 7 squared is 49; that is, 72=(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)= (-3)(-3)=9 but -3= -9. Exponents can also be negative or zero; such exponents are defined as follows.

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

Negative exponents: For all nonzero numbers a, a-1 = 1/a, a-2 = 1/a2 , a-3 = 1/a3 and so on. Note that (a)(a-1) = (a)(1/a) = 1


A square root of a nonnegative number n is a number r such that r= n. For example, 4 is a square root of 16 because 4=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.