Simplifying Radicals Unit 10 Lesson 2 Simplifying Radicals Students will be able to: β’ Use square root and cube root symbols to represent solutions to equations of the form π π = π and π π = π, where p is a positive rational number β’ Simplify expressions involving radicals using the properties of radicals. Simplifying Radicals Key Vocabulary: β’ Radical β’ Radicand β’ Simplifying Radicals β’ Denominator β’ Like Radicals Simplifying Radicals Radicals (or roots) are the opposite operation of applying exponents. A power can be undone with a radical and a radical can be undone with a power. π β’ π = π ππ π β’ π = π ππ π=π π π=π πππ ππ πππ πππππ Radical sign Simplifying Radicals Common Radicals: β’ A square (second) root is written as β β’ A cube (third) root is written as β β’ A fourth root is written as β Simplifying Radicals β’ To simplify a radical, factor the expression under the radical sign to its prime factors. β’ For every pair of like factors, bring out one of the factors. β’ Multiply whatever is outside the sign, and then multiply whatever is inside the sign. β’ Remember that for each pair, you βbring outβ only one of the numbers. Simplifying Radicals β’ Variables in a radicand are simplified in the same way. You've got a pair of can be taken "out front". β’ Negative Radicals The only restriction that exists for negative signs and radicals is that there cannot be a negative sign under an even root since there is no real solution to this problem. However, a negative sign can exist in front of a radical or under odd roots and still be able to obtain a real number. Simplifying Radicals β’ PRODUCT RULE FOR RADICAL The radical of the product is the product of two radicals. π ππ = π πβπ π Simplifying Radicals QUOTIENT RULE FOR RADICAL The radical of a quotient is the quotient of the radicals. π π = π π π π π πβ π Simplifying Radicals Sample Problem 1: Simplify the following expressions. Assume that all variables represent positive real numbers. π. ππ = π. ππππ = π. ππ = Simplifying Radicals Sample Problem 1: Simplify the following expressions. Assume that all variables represent positive real numbers. π. ππ = π β π = π. ππππ = π. ππ = ππ β π = π π π β ππ β ππ = π β ππ β ππ = ππ π ππ β π = ππ β π = π π Simplifying Radicals Sample Problem 1: Simplify the following radicals. Assume that all variables represent positive real numbers. ππ ππ d. e. f. π π = βππππ πππ πππ ππ = = Simplifying Radicals Sample Problem 1: Simplify the following radicals. Assume that all variables represent positive real numbers. ππ ππ d. e. f. π π = ππ ππ πππ ππ = = π βππππ πππ πβπ πβπ = π π πππ π ππ = βπ π βππ π ππ β ππ π = ππ ππ =± βππ ππ = π πβ π π β ππ π π ππ β π π ππ = π π ππ ππ Simplifying Radicals β’ Radicals can also be written in exponent notation. However, in this case the exponent would be a fraction. π π π ππ = π π ππ = π β’ If π is odd then β’ If π is even then π ππ = π Simplifying Radicals Sample Problem 2: Write each expression in radical notation. a. π π π = b. βπ π π c. π = π π = Simplifying Radicals Sample Problem 2: Write each expression in radical notation. a. π π π b. βπ π π c. π = = π = ±π π π π = π π βπ = π βπ π = βπ Simplifying Radicals Sample Problem 3: Write each expression in exponential notation. a. b. π π π βππ = ππ = c. πππ = Simplifying Radicals Sample Problem 2: Write each expression in exponential notation. a. π βππ = βππ π π πβ π π π b. ππ = π c. πππ = π π π = βπ =π π π ππ = ±π = ±π πβπ = βπ Simplifying Radicals RATIONALIZING THE DENOMINATOR When dealing with fractions, a final answer cannot contain radicals in the denominator. Therefore, it is necessary to eliminate any radical from the denominator. The process of removing the radical from the denominator is called rationalizing the denominator. Simplifying Radicals Rationalizing the denominator 1. An expression is considered simplified only if there is no radical sign in the denominator. π π ππ = π π ππ π βπ ππβπ ππβπ π = ππβπ π 2. Rationalizing the denominator with two terms, one or both of which involve square root. Step 1: Multiply the numerator and denominator by the conjugate of the denominator Step 2: Simplify the resulting expression if possible Simplifying Radicals Sample Problem 4: Simplify the following expressions. Assume that all variables represent positive real numbers. a. b. 7 5 = 2 π₯+π¦ = Simplifying Radicals Sample Problem 4: Simplify the following expressions. Assume that all variables represent positive real numbers. a. b. π π = π π+π π π β = π π = π π+π π π π β π+π π+π = π π+π π+π Simplifying Radicals Sample Problem 4: Simplify the following expressions. Assume that all variables represent positive real numbers. c. d. π πβ π π πβ π = = Simplifying Radicals Sample Problem 4: Simplify the following expressions. Assume that all variables represent positive real numbers. c. π πβ π π d. πβ π = π πβ π = π πβ π β π+ π π+ π β π+ π π+ π = = π π+ π ππ β π π π π+ π ππ β π π = = π π+ π βπ π π+ π πβπ =β π+ π Simplifying Radicals β’ POWER OF A RADICAL The radicand has to be raised to the exponent of the power while the index remains the same. π ππ π = π ππβπ Simplifying Radicals β’ RADICAL OF A RADICAL The radicand stays the same, and the index is the product of the indices. π π π = πβπ π Simplifying Radicals Sample Problem 5: Simplify the following expressions. Assume that all variables represent positive real numbers. π a. π b. πππ π ππ = = Simplifying Radicals Sample Problem 5: Simplify the following expressions. Assume that all variables represent positive real numbers. π a. π b. πππ ππ π = = πβπ π ππβπ ππ = = π π ππ ππ π =π π Simplifying Radicals Like radicals are radicals that have the same index and the same radicand Simplifying Radicals Sample Problem 6: Simplify radicals and recognize like or unlike radicals. a. π π ππ§π π π b. πππ ; π ππ ππ§π ππ Simplifying Radicals Sample Problem 6: Simplify radicals and recognize like or unlike radicals. π. π π πππ π π π³π°π²π¬ πΉπ¨π«π°πͺπ¨π³πΊ b. πππ ; π ππ πππ ππ ππ β π; π ππ β π; πππ π β π π π; ππ π; πππ π π π³π°π²π¬ πΉπ¨π«π°πͺπ¨π³πΊ Simplifying Radicals Sample Problem 6: Simplify radicals and recognize like or unlike radicals. π c. π ππ πππ π ππ d. π πππ ; ππ π; ππππ Simplifying Radicals Sample Problem 6: Simplify radicals and recognize like or unlike radicals. π c. π ππ πππ π ππ πΌπ΅π³π°π²π¬ πΉπ¨π«π°πͺπ¨π³πΊ d. π πππ ; ππ π; ππππ ππ π ; ππ π; ππ π π³π°π²π¬ πΉπ¨π«π°πͺπ¨π³πΊ
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