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10.3 Simplifying Radical Expressions

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Use the product rule for radicals. Objective 1 Slide 10.3- 2

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Product Rule for Radicals If and are real numbers and n is a natural number, then That is, the product of two nth roots is the nth root of the product. Slide 10.3- 3 Use the product rule for radicals. Use the product rule only when the radicals have the same index.

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Multiply. Assume that all variables represent positive real numbers. Slide 10.3- 4 CLASSROOM EXAMPLE 1 Using the Product Rule Solution:

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Multiply. Assume that all variables represent positive real numbers. This expression cannot be simplified by using the product rule. Slide 10.3- 5 CLASSROOM EXAMPLE 2 Using the Product Rule Solution:

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Use the quotient rule for radicals. Objective 2 Slide 10.3- 6

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Quotient Rule for Radicals If and are real numbers, b ≠ 0, and n is a natural number, then That is, the nth root of a quotient is the quotient of the nth roots. Slide 10.3- 7 Use the quotient rule for radicals.

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Simplify. Assume that all variables represent positive real numbers. Slide 10.3- 8 CLASSROOM EXAMPLE 3 Using the Quotient Rule Solution:

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Simplify radicals. Objective 3 Slide 10.3- 9

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Conditions for a Simplified Radical 1. The radicand has no factor raised to a power greater than or equal to the index. 2. The radicand has no fractions. 3. No denominator contains a radical. 4. Exponents in the radicand and the index of the radical have greatest common factor 1. Slide 10.3- 10 Simplify radicals. Be careful with which factors belong outside the radical sign and which belong inside.

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Simplify. Cannot be simplified further. Slide 10.3- 11 CLASSROOM EXAMPLE 4 Simplifying Roots of Numbers Solution:

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Simplify. Assume that all variables represent positive real numbers. Slide 10.3- 12 CLASSROOM EXAMPLE 5 Simplifying Radicals Involving Variables Solution:

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Simplify. Assume that all variables represent positive real numbers. Slide 10.3- 13 CLASSROOM EXAMPLE 6 Simplifying Radicals by Using Smaller Indexes Solution:

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If m is an integer, n and k are natural numbers, and all indicated roots exist, then Slide 10.3- 14 Simplify radicals.

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Simplify products and quotients of radicals with different indexes. Objective 4 Slide 10.3- 15

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Simplify. The indexes, 2 and 3, have a least common index of 6, use rational exponents to write each radical as a sixth root. Slide 10.3- 16 CLASSROOM EXAMPLE 7 Multiplying Radicals with Different Indexes Solution:

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Use the Pythagorean theorem. Objective 5 Slide 10.3- 17

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Pythagorean Theorem If c is the length of the longest side of a right triangle and a and b are lengths of the shorter sides, then c 2 = a 2 + b 2. The longest side is the hypotenuse, and the two shorter sides are the legs, of the triangle. The hypotenuse is the side opposite the right angle. Slide 10.3- 18 Use the Pythagorean theorem.

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Find the length of the unknown side of the triangle. c 2 = a 2 + b 2 c 2 = 14 2 + 8 2 c 2 = 196 + 64 c 2 = 260 The length of the hypotenuse is Slide 10.3- 19 CLASSROOM EXAMPLE 8 Using the Pythagorean Theorem Solution:

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Find the length of the unknown side of the triangle. c 2 = a 2 + b 2 6 2 = 4 2 + b 2 36 = 16 + b 2 20 = b 2 The length of the leg is Slide 10.3- 20 CLASSROOM EXAMPLE 8 Using the Pythagorean Theorem (cont’d) Solution:

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Use the distance formula. Objective 6 Slide 10.3- 21

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Distance Formula The distance between points (x 1, y 1 ) and (x 2, y 2 ) is Slide 10.3- 22 Use the distance formula.

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Find the distance between each pair of points. (2, –1) and (5, 3) Designate which points are (x 1, y 1 ) and (x 2, y 2 ). (x 1, y 1 ) = (2, –1) and (x 2, y 2 ) = (5, 3) Start with the x- value and y-value of the same point. Slide 10.3- 23 CLASSROOM EXAMPLE 9 Using the Distance Formula Solution:

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Find the distance between each pairs of points. (–3, 2) and (0, 4) Designate which points are (x 1, y 1 ) and (x 2, y 2 ). (x 1, y 1 ) = (–3, 2) and (x 2, y 2 ) = (0, 4) Slide 10.3- 24 CLASSROOM EXAMPLE 9 Using the Distance Formula (cont’d) Solution:

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