Gre math Review 1 Arithmetic

1.3 Exponents and Roots

Exponents are used to denote the repeated multiplication of a number by itself; for example,   3 superscript 4 = 3 times 3 times 3 times 3, that is 3 multiplied by itself 4 times, which is equal to 81, and 5 superscript 3 = 5 times 5 times 5, that is 5 multiplied by itself 3 times, which is equal to 125.

In the expression   3 superscript 4, 3 is called the base, 4 is called the exponent, and we read the expression as “3 to the fourth power.” So 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 squared = 6 times 6 = 36, and 7 squared is 49; that is,   7 squared = 7 times 7 = 49.

When negative numbers are raised to powers, the result may be positive or negative. For example,   open parenthesis, negative 3, close parenthesis, squared = negative 3 times negative 3, which is equal to 9,
while   open parenthesis, negative 3, close parenthesis, to the fifth power = negative 3 multiplied by itself 5 times, which is equal to negative 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 without the parentheses, the expression   negative 3 squared means the negative of 3 squared; that is, the exponent is applied before the negative sign. So   open parenthesis, negative 3, close parenthesis, squared = 9, but negative 3 squared = negative 9.

Exponents can also be negative or zero; such exponents are defined as follows.

The exponent zero: For all nonzero numbers a,
  a to the power 0 = 1.
The expression   0 to the power 0 is undefined.

Negative exponents: For all nonzero numbers a,


a to the power negative 1 = 1 over a, a to the power negative 2 = 1 over a squared, a to the power negative 3 = 1 over a to the third power, etc.
Note that  
a, times, a to the power negative 1 = a, times, 1 over a, which is equal to 1.

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

Another square root of 16 is   negative 4, since  open parenthesis, negative 4, close parenthesis, squared = 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,   the square root symbol over the number 100 = 10, a minus sign followed by, the square root symbol over the number 100 = negative 10, and the square root symbol over the number 0 = 0.
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 is greater than 0 and b is greater than 0.

Rule 1:   open parenthesis, the positive square root of a, close parenthesis, squared = a
Example A:   open parenthesis, the positive square root of 3, close parenthesis, squared = 3
Example B:   open parenthesis, the positive square root of pi, close parenthesis, squared = pi

Rule 2:   the positive square root of, a squared = a

Example A:   the positive square root of 4 = 2
Example B:   the positive square root of, pi squared = pi

Rule 3:   the positive square root of a times the positive square root of b = the positive square root of a b

Example A:   the positive square root of 3 times the positive square root of 10 = the positive square root of 30
Example B:   the positive square root of 24 = the positive square root of 4 times the positive square root of 6, which is equal to 2 times the positive square root of 6

Rule 4:   the positive square root of a over the positive square root of b = the positive square root of the fraction a over b

Example A:   the positive square root of 5 over the positive square root of 15 = the positive square root of the fraction 5 over 15, which is equal to the positive square root of the fraction 1 over 3
Example B:  the positive square root of 18 over the positive square root of 2 = the positive square root of the fraction 18 over 2, which is equal to the positive square root of 9, or 3

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 and fourth root of n 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,   the cube root of 8 = 2, but 8 has two fourth roots:   the positive fourth root of 8 and the negative fourth root of 8; and   negative 8 has exactly one cube root,   the cube root of negative 8 = negative 2, but   negative 8 has no fourth root, since it is negative.