Ch 8 - Rational & Radical Functions Simplifying Radical Expressions

Product Property of Radicals: For any real numbers a and b, then

Simplify.

Quotient Property of Radicals For any real numbers a and b, and b ≠ 0, if all roots are defined. To rationalize the denominator, you must multiply the numerator and denominator by a quantity so that the radicand has an exact root.

Simplify.

Two radical expressions are called like radical expressions if the radicands are alike. Conjugates are binomials in the form and where a, b, c, and d are rational numbers. The product of conjugates is always a rational number.

Simplify.

nth roots The nth root of a real number a can be written as the radical expression, where n is the index of the radical and a is the radicand.

Numbers and Types of Real Roots CaseRootsExample Odd Index1 real rootThe real 3 rd root of 8 is 2. The real 3 rd root of -8 is -2. Even Index, Positive Radicand 2 real rootsThe real 4 th roots of 16 are ±2. Even Index, Negative Radicand 0 real roots-16 has no real 4 th roots. Radicand of 01 root of 0The 3 rd root of 0 is 0.

Find all real roots. A. Sixth roots of 64 B. Cube roots of -216 C. Fourth roots of -1024

Properties of nth Roots Product Property of Roots Quotient Property of Roots

Simplify each expression. Assume that all variables are positive.

A rational exponent is an exponent that can be expressed as m/n, where m and n are integers and n ≠ 0. The exponent 1/n indicates the nth root. The exponent m/n indicates the nth root raised to the mth power.

Write each expression by using rational exponents. Write each expression in radical form and simplify.

Properties of Rational Exponents Product of Powers Property Quotient of Powers Property Power of a Power Property Power of a Product Property Power of a Quotient Property

Simplify each expression.