Chapter R Section 7: Radical Notation and Rational Exponents In this section, we will… Evaluate nth Roots Simplify Radical Expressions Add, Subtract, Multiply and Divide Radical Expressions Rationalize Denominators Simplify Expressions with Rational Exponents Factor Expressions with Radicals or Rational Exponents

Recall from Review Section 2… If a is a non-negative real number, any number b, such that is the square root of a and is denoted If a is a non-negative real number, any non-negative number b, such that is the primary square root of a and is denoted Examples: Evaluate the following by taking the square root. The principal root of a positive number is positive principal root: Negative numbers do not have real # square roots

where a, b are any real number if n is odd The principal nth root of a real number a, n > 2 an integer, symbolized by is defined as follows: where and if n is even where a, b are any real number if n is odd Examples: Simplify each expression. index radicand radical principal root:

Properties of Radicals: Let and denote positive integers and let a and b represent real numbers. Assuming that all radicals are defined: Simplifying Radicals: A radical is in simplest form when: No radicals appear in the denominator of a fraction The radicand cannot have any factors that are perfect roots (given the index) Examples: Simplify each expression.

Simplifying Radical Expressions Containing Variables: Examples: Simplify each expression. Assume that all variables are positive. When we divide the exponent by the index, the remainder remains under the radical

Adding and Subtracting Radical Expressions: simplify each radical expression combine all like-radicals (combine the coefficients and keep the common radical) Examples: Simplify each expression. Assume that all variables are positive.

Multiplying and Dividing Radical Expressions: Examples: Simplify each expression. Assume that all variables are positive.

Examples: Simplify each expression Examples: Simplify each expression. Assume that all variables are positive.

Assignment Page 52 Problems 26 - 58 evens and #59

Rationalizing Denominators: Recall that simplifying a radical expression means that no radicals appear in the denominator of a fraction. Examples: Simplify each expression. Assume that all variables are positive.

Rationalizing Binomial Denominators: example: Examples: Simplify each expression. Assume that all variables are positive. The conjugate of the binomial a + b is a – b and the conjugate of a – b is a + b.

Assignment Page 53 Problems 66 – 86 evens

Evaluating Rational Exponents: Examples: Simplify each expression.

Convert between radical and exponential notation… Examples: y5/6 = Whenever you have an integer, try typing the problem into your calculator first. 163/4 = 8  m5/3n7/3 = = 173/5 = 5 1/2 • 5 1/3 = 5 5/6

Simplifying Expressions Containing Rational Exponents: Recall the following from Review Section 2: Laws of Exponents: For any integers m, n (assuming no divisions by 0) and new! new!

Examples: Simplify each expression Examples: Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive.