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Math 10Lesson 1-5 Mixed and Entire RadicalsI.Entire and mixed radicalsAn entire radical is a number in a radical with no coefficient or multiplying number in front ofthe radical.32342000162are all examples of entire radicals.A mixed radical is a number in a radical with a coefficient or multiplying number in front of theradical.4 23164 3 200087 4 162are all examples of mixed radicals.II. Multiplication of radicalsIf the radicand of a radical (i.e. – the number inside the radical) can be factored, we can expressthe original radical as a multiplication of the two factors. For example, consider 35 . 35 can befactored into 5·7. Therefore we can write35 5 7 5 7In the opposite sense, if the index is the same for both radicals, we can combine two radicalsinto one radical49 4 11 4 9 11 4 99In general, if a and b are real numbers and n is a natural number,nab n a n b n a n bDr. Ron LichtL1–5 Mixed and entire radicals1www.structuredindependentlearning.com

III. Converting/simplifying an entire radical into a mixed radicalTo convert an entire radical into a mixed radical we need to find perfect squares or perfectcubes or perfect quartics, depending on the index, that are factors of our original radicand. Ifthere are such factors we can then remove them from the radicand.Consider, for example, 20 . First, we make a list of perfect squares: 22 4, 32 9, 42 16,and so on. The largest perfect square factor of 20 is 4. Therefore20 4 5but note that4 2 . Therefore we can write20 2 5This is how we convert an entire radical into a mixed radical and 2 5 is referred to as thesimplified form of20 .As demonstrated in the example above, the basic process for simplifying a radical ( n x ) is tolook for factors of x that are perfect squares (if n 2), perfect cubes (if n 3), a perfect quartic(if n 4), etc. and then remove them from the radical. Perhaps a few examples may help.Example 180 .SimplifyFor square roots, we look for the largest factor that is perfect square (i.e. 4, 9, 16, 25 ). The numbers 4 and 16 are factors of 80, so we choose the largest: 16 580 16 5 4 5Example 2Simplify3144 .For cube roots (n 3), we look for factors of 144 that are perfect cubes (i.e. 23 8,33 27, 43 64, 53 125 ). We try 8, 27 and 64. 144 8 18, 144 27 5.33and 144 64 2.25 . Therefore we use 8 and 18.3144 3 8 18 3 8 3 18 2 3 18Dr. Ron LichtL1–5 Mixed and entire radicals2www.structuredindependentlearning.com

Example 3Simplify 4 162 .In general, the larger the index the harder it is to spot appropriate factors to try. Adifferent strategy from the one used in Examples 1 and 2 is to factor the radicand intoits prime factors. Since we are dealing with the 4th root, we are looking for prime factorsthat repeat 4 times. The prime factorization of 162 is 2·3·3·3·3 or 2·34 2·81. Thus,4162 4 2 81 4 2 4 81 34 2Example 4Simplify 162 98Note that we cannot add 162 and 98 together: 162 98 162 98There is no obvious way to simplify this expression. Thus, we try to simply each radicaland then see what we can do from there.162 98 81 2 49 2 9 2 7 2 16 2Note, the Pearson text book uses prime factorization as its prime strategy. While it is a goodstrategy to use at times, it is not wise to use it as your only strategy when solving problems. Itis wise to approach problem solving with a more playful and open attitude that is capable ofseeing new and creative ways to solve problems. A playful, creative approach also has thebenefit of removing some of the boredom and tedium of some math assignments.Question 1Simplify the following:754Dr. Ron LichtL1–5 Mixed and entire learning.com

IV. Expanding mixed radicals into entire radicalsExpanding a mixed radical into an entire radical is the reverse process of simplifying radicals.Generally speaking it is an easier process.Example 5Write 4 3 as an entire radical.Since the index of the radical is 2, we raise 4 to the 2nd power (42) and then multiply thisby the radicand.4 3 42 3 16 3 48Example 6Write 2 5 6 as an entire radical.Since the index of the radical is 5, we raise 2 to the 5th power (25) and then multiply thisinto the radical.25 6 5 25 5 6 5 32 5 6 5 192Example 733 27Write 3 3 2 as an entire radical. 3 3 2 3 27 3 2 3 54Question 2Write the following as entire radicals.3 736 725 8Dr. Ron LichtL1–5 Mixed and entire radicals4www.structuredindependentlearning.com

V. Nasty question of the dayExample 83 1.732 and without using a calculator determine a decimalapproximation for 12 .Given thatPerhaps we could change12 4 3Since12 into a mixed radical and then see where that leads us. 2 31.7323 2 3 2 1.732 3.464Therefore 12 3.46412Given that 2 1.4142 and without using a calculator determine a decimal approximationfor each radical.(a)200(b)20000(c)8(d)18(e)32(f)50Dr. Ron LichtL1–5 Mixed and entire radicals5www.structuredindependentlearning.com

VI. Assignment1.Write each radical in simplest form.a) 8b) 12c) 32e)18f)27g)48h)2.Write each mixed radical as an entire radical.a) 5 2b) 6 2e) 5 3f) 6 33.Write each radical in simplest form, if possible.d) 600e) 54f) 91g)4.5.28h)335075d)i) 112Write each radical in simplest form, if possible.a) 3 16c) 3 256e) 3 60g)3135Write each mixed radical as an entire radical.a) 3 2c) 6 5e) 7 7g) 3 3 3i)3500i) 5 3 26.Express the side length of this 252 ft2 square as a radical in simplest form.7.A cube has a volume of 200 cm3. Write the edge length of the cube as aradical in simplest form.8.A square has an area of 54 square inches. Determine the perimeter of the square. Writethe answer as a radical in simplest form.9.Write each radical in simplest form.a) 4 48c)4125010. Write each mixed radical as an entire radical.a) 6 4 3c)35 4Dr. Ron LichtL1–5 Mixed and entire radicals6www.structuredindependentlearning.com

11. Here is a student’s solution for writing as an entire radical.83 2 8 3 2 3 2 3 2 3 2 2 34Identify an error the student made, then write the correct solution.12. A student simplified a radical as shown:96 4 48 2 48 2 8 6 2 4 6 8 6Identify the errors the student made, then write a correct solution.13. Simplify the radicals in each list. What patterns do you see in the results? Write the next 2radicals in each list.a)4c)8400800400008000014. The largest square in this diagram has side length 8 cm. Calculatethe side length and area of each of the two smaller squares.Write the radicals in simplest form.Dr. Ron LichtL1–5 Mixed and entire radicals7www.structuredindependentlearning.com

Dr. Ron Licht 1 www.structuredindependentlearning.com L1–5 Mixed and entire radicals Math 10 Lesson 1-5 Mixed and Entire Radicals I. Entire and mixed radicals An entire radical is a number in a radical with no coefficient or multiplying number in front of the radical. 23 3 2000 4 162 are all examples of entire radicals. A mixed radical is a number in a radical with a coefficient or .