1. Properties of Positive Integer Exponents, m > 0, n > 0 for example,

2. More Properties of Positive Integer Exponents, m > 0, n > 0 for example,

3. Properties of Integer Exponents, a 0 for example,

4. Scientific Notation A number is written in scientific notation if it is of the form where 1 a < 10, and m is some integer. Scientific notation is useful for writing very large or very small numbers since fewer zeros are required in the representation of the number. Example. One light-year is about 6 trillion miles, which is 6,000,000,000,000 miles using decimal notation. This may be written as using scientific notation.

5. More about Scientific Notation If the result of a measurement or calculation is written in scientific notation as where 1 a < 10, and m is some integer, then the number of digits of a are taken as the significant digits of the result. If we write a number in scientific notation with fewer significant digits than the original number presented, we must round the last significant digit according to the rule:  add 1 to last significant digit if digit following it in the original number is 5, 6, 7, 8, or 9  leave the last significant digit alone if digit following it in the original number is 0, 1, 2, 3, or 4 Example. The speed of light is 186,282 miles per second. This is 1.863  10 5 to four significant digits.

6. Summary of Integer Exponents; We discussed Six properties for positive integer exponents Three properties for general integer exponents Scientific notation Significant digits

7. Rational Exponents and Radicals If n is a natural number and n is odd, b 1/n can be defined to be the unique nth root of b. If n is a natural number and n is even, b 1/n can be defined to be the unique positive nth root of b. In this case, we say that b 1/n is the principal nth root of b. Now, we define b m/n for an integer m, a natural number n, and a real number b, by where b must be positive when n is even. With this definition, all the rules of exponents continue to hold when the exponents are rational numbers.

8. Examples for Rational Exponents Simplify 27 4/3. We have Simplify We have Simplify. We have Simplify 16 3/4. We have Note that 16 1/4 is taken to be the principal root, which is 2.

9. Radicals The principal nth root of a real number b was discussed previously. An alternative representation of this root is available using a radical symbol,. We summarize as follows. with these restrictions:  if n is even and b < 0, is not a real number,  if n is even and b 0, is the nonnegative number a satisfying a n = b. Warning.

10. Radicals and Exponents We have that Example. Problem. Change from rational exponent form to radical form, Assume that x and y are positive real numbers. Solution.

11. Properties of Radicals for example,

12. More about Radicals For example, Simplify

13. Rationalizing Denominators A fraction is sometimes considered simplified if its denominator is free of radicals. The process by which this is accomplished is called rationalizing the denominator. In this connection, a useful formula is: Example. Rationalize the denominator:

14. Examples for Simplifying Expressions Involving Radicals.

15. Rational Expressions and Radicals; We discussed nth roots and the principal nth root rational exponents radicals converting from rational exponent form to radical form and vice versa properties of radicals rationalizing the denominator