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Ch 8 - Rational & Radical Functions Simplifying Radical Expressions

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

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Simplify.

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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.

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Simplify.

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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.

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Simplify.

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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.

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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.

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

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

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Simplify each expression. Assume that all variables are positive.

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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.

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

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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

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Simplify each expression.

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