252 CHAPTER 5 Number Theory and the Real Number System To simplify a radical, write the radical as a product of two radicals. One of the radicals should contain the greatest perfect square that is a factor of the radicand in the original expression. Then simplify the radical containing the perfect square factor. For example, 12 43 4 32323 = ⋅ = ⋅ = ⋅ = and 75 253 2535353 = ⋅ = ⋅ = ⋅ = MATHEMATICS TODAY A Piece of the Pi The ratio of a circle’s circumference to its diameter is the constant we know as pi, .π Like any irrational number, π has a decimal representation that is a nonterminating, nonrepeating decimal number. Computing the digits of π has been of interest to mathematicians for hundreds of years, since Sir Isaac Newton developed formulas to calculate the digits. In March 2022, Emma Haruka Iwao of Google determined pi to 100 trillion or 100,000,000,000,000 digits. The calculation, using over 25 machines took 157 days to accomplish. This new record surpassed the old record, set in 2019, by more than 68 trillion digits. Why This Is Important Computations to determine a value to any number of decimal places are part of a branch of mathematics called arbitrary-precision arithmetic . Records such as digits of ,π and the largest prime number, will continually be broken as mathematicians and computer scientists improve computers and the formulas used to find them. C C 5 pd or p 5 C d d r Example 1 Simplifying Radicals Simplify. a) 27 b) 32 c) 75 Solution a) Since 9 is a perfect square factor of 27, we write 27 93 9 33333 = ⋅ = ⋅ = ⋅ = Since 3 has no perfect square factors, 3 cannot be simplified. b) Since 16 is a perfect square factor of 32, we write 32 162 16 2 42 = ⋅ = ⋅ = c) Since 25 is a perfect square factor of 75, we write 75 253 25 3 53 = ⋅ = ⋅ = 7 Now try Exercise 37 In Example 1(b), you can obtain the correct answer if you start out factoring differently: 32 4 8 4 8 2 8 = ⋅ = ⋅ = ⋅ Note that 8 has 4 as a perfect square factor. 2824224 222242 = ⋅ = ⋅ ⋅ = ⋅ ⋅ = The second method will eventually give the same answer, but it requires more work. It is best to try to factor out the largest perfect square factor from the radicand. Addition and Subtraction of Radicals To add or subtract two or more square roots with the same radicand, add or subtract their coefficients while keeping the common radicand. The answer is the sum or difference of the coefficients multiplied by the common radical. Example 2 Adding and Subtracting Radicals with the Same Radicand Simplify. a) 8 3 5 3 + b) 7 7 12 7 7 − + Solution a)83 53 (8 5)3 133 + = + = b)77 127 7 (7 12 1)7 47 − + = − + = − Note that = 7 1 7. 7 Now try Exercise 45
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