Exponents and Radicals Objective: To review rules and properties of exponents and radicals.

Exponential Notation

Properties of Exponents

Example 1 Use the properties of exponents to simplify each expression. a)

Example 1 Use the properties of exponents to simplify each expression. a)

Example 1 Use the properties of exponents to simplify each expression. You Try: b)

Example 1 Use the properties of exponents to simplify each expression. You Try: b)

Example 1 Use the properties of exponents to simplify each expression. You Try: c)

Example 1 Use the properties of exponents to simplify each expression. You Try: c)

Example 1 Use the properties of exponents to simplify each expression. You Try: d)

Example 1 Use the properties of exponents to simplify each expression. You Try: d)

Example 2 Rewrite each expression with positive exponents. a)

Example 2 Rewrite each expression with positive exponents. a)

Example 2 Rewrite each expression with positive exponents. b)

Example 2 Rewrite each expression with positive exponents. b)

Example 2 Rewrite each expression with positive exponents. You Try: c)

Example 2 Rewrite each expression with positive exponents. You Try: c)

Example 2 Rewrite each expression with positive exponents. You Try: d)

Example 2 Rewrite each expression with positive exponents. You Try: d)

Radicals and Their Properties Definition of nth Root of a Number. Let a and b be real numbers and let n > 2 be a positive integer. If a = b n then b is an nth root of a. If n = 2, the root is a square root. If n = 3, the root is a cube root.

Radicals and Their Properties Principal nth Root of a Number. Let a be a real number that has at least one nth root. The principal nth root of a is the nth root that has the same sign as a. It is denoted by a radical symbol The positive integer n is the index of the radical, and the number a is the radicand. If n = 2, omit the index and write

Example 5 Evaluate: a)

Example 5 Evaluate: a)

Example 5 Evaluate: b)

Example 5 Evaluate: b)

Example 5 Evaluate: c)

Example 5 Evaluate: c)

Example 5 Evaluate: d)

Example 5 Evaluate: d)

Example 5 Evaluate: You Try: d)

Example 5 Evaluate: You Try: d)

Example 5 Evaluate: e)

Example 5 Evaluate: e)

Properties of Radicals

Example 6 Use the properties of radical to simplify each expression. a)

Example 6 Use the properties of radical to simplify each expression. a)

Example 6 Use the properties of radical to simplify each expression. b)

Example 6 Use the properties of radical to simplify each expression. b)

Example 6 Use the properties of radical to simplify each expression. c)

Example 6 Use the properties of radical to simplify each expression. c)

Example 6 Use the properties of radical to simplify each expression. d)

Simplifying Radicals An expression involving radicals is in simplest form when the following conditions are satisfied. 1.All possible factors have been removed from the radical. 2.All fractions have radical-free denominators (accomplished by a process called rationalizing the denominator). 3.The index of the radical is reduced.

Example 7 Simplify each radical. a)

Example 7 Simplify each radical. a)

Example 7 Simplify each radical. You Try: a)

Example 7 Simplify each radical. You Try: a)

Example 7 Simplify each radical. b)

Example 7 Simplify each radical. b)

Example 7 Simplify each radical. c)

Example 7 Simplify each radical. c)

Example 7 Simplify each radical. You Try: c)

Example 7 Simplify each radical. You Try: c)

Example 8 Simplify each radical. a)

Example 8 Simplify each radical. a)

Example 8 Simplify each radical. b)

Example 8 Simplify each radical. b)

Example 8 Simplify each radical. You Try: c)

Example 8 Simplify each radical. You Try: c)

Example 9 Combine each radical. a)

Example 9 Combine each radical. a)

Example 9 Combine each radical. You Try: b)

Example 9 Combine each radical. You Try: b)

Example 10 Rationalize the denominator of each expression. a)

Example 10 Rationalize the denominator of each expression. a)

You Try Rationalize the denominator of each expression. You Try: b)

Example 10 Rationalize the denominator of each expression. You Try: b)

Example 11 Rationalize the denominator of each expression. a)

You Try Rationalize the denominator of each expression. You Try: b)

You Try Rationalize the denominator of each expression. You Try: b)

Rational Exponents The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken.

Example 13 Change the base from radical to exponential form. a)

Example 13 Change the base from radical to exponential form. a)

Example 13 Change the base from radical to exponential form. b)

Example 13 Change the base from radical to exponential form. b)

You Try Change the base from radical to exponential form. You Try: c)

You Try Change the base from radical to exponential form. You Try: c)

Example 14 Change the base from exponential to radical form. a)

Example 14 Change the base from exponential to radical form. a)

You Try Change the base from exponential to radical form. b)

You Try Change the base from exponential to radical form. b)

Example 15 Simplify each rational expression. a)

Example 15 Simplify each rational expression. a)

Example 15 Simplify each rational expression. You Try: b)

Example 15 Simplify each rational expression. You Try: b)

You Try Simplify each rational expression. You Try: c)

You Try Simplify each rational expression. You Try: c)

You Try Simplify each rational expression. You Try: d)

You Try Simplify each rational expression. You Try: d)

Homework Pages odd odd multiples of 3