Exponents and Radicals - Radicals Notes

introduction

Math 10C - RadicalsRadicals Math Animations
Transcript of: Radicals.

Radicals: Introduction

 

  1. In this lesson, we will learn about radicals.
  2. In part a of the introduction, we will label the parts of a radical.
  3. The entire expression is called a radical.
  4. The shape that resembles a checkmark is called the radical symbol.
  5. The number inside the radical symbol is called the radicand.
  6. And the number squeezed between the checkmark is called the index.
  7. Moving on to part b, we are asked "What is the index of root(5)?"
  8. The index of this radical is 2. All square roots have an index of 2, but for convenience we don't bother writing it in.
  9. Moving on to part c, what is the difference between an entire radical and a mixed radical?
  10. An entire radical does not have a coefficient. The examples shown are all entire radicals because they lack a coefficient.
  11. A mixed radical has a coefficient. The examples shown are all mixed radicals because they have a coefficient.
  12. In part d, we are asked if it's possible to write a radical without using the radical symbol.
  13. Yes it is possible. Radicals can be expressed as fractional exponents. The examples show how a radical can be written with a fractional exponent. We'll learn more about this topic later in the lesson.

 

Example 1: Converting Entire Radicals to Mixed Radicals.

 

  1. Convert each of the following entire radicals to mixed radicals. In part a, we'll convert the square root of 20 to a mixed radical.
  2. The first method we'll learn is called the prime factorization method.
  3. Start with root(20).
  4. Now find any two numbers that multiply to 20. 2 times 10 will work.
  5. 10 can be broken down to 2 times 5.
  6. Now that we've written 20 as 2 × 2 × 5, we can't break things down any further since all of the numbers are primes.
  7. Next, we must identify pairs of identical numbers. There is one pair of 2's.
  8. Now pull out the pair of 2's. They collapse to a single 2 on the outside of the radical. The answer is 2root(5).
  9. The second method we can use to find the mixed radical is called the perfect square method.
  10. Start with root(20).
  11. Write 20 as a product of two numbers, one of which is a perfect square.
  12. Split the radical.
  13. The square root of 4 is 2. The square root of 5 is an irrational number, so we'll just leave this as root(5). The answer is 2root(5).
  14. Moving on to part b, we'll convert root(32) to a mixed radical.
  15. First we'll convert this to a mixed radical using the prime factorization method.
  16. Start with root(32).
  17. Find any two numbers that multiply to 32. We'll use 4×8.
  18. 4 can be broken down to 2 × 2, and 8 can be broken down to 2 × 2 × 2. We now have broken down 32 into its prime factors.
  19. There are 2 pairs of 2's.
  20. Each pair of 2's collapses to a single 2 on the outside of the radical. The 2's outside the radical are connected with multiplication.
  21. Simplify to get the answer 4root(2).
  22. Now we'll try this again, using the perfect square method.
  23. Start with root(32).
  24. Find two numbers that multiply to 32, one of which is a perfect square. We'll use 16 times 2.
  25. Split the radical.
  26. The square root of 16 is 4. The square root of 2 is an irrational number, so we'll leave this as root(2).
  27. In part c, we'll convert the cube root of 16 to a mixed radical.
  28. First we'll convert this to a mixed radical using the prime factorization method.
  29. Start with the cube root of 16.
  30. Find two numbers that multiply to 16. 4 times 4 will work.
  31. Each 4 can be broken down to 2 × 2. We have now expressed 16 as a product of its prime factors.
  32. Since the index is 3, we need to find a set of three identical numbers.
  33. The three 2's collapse to a single two outside the radical. The answer is 2 times the cube root of 2.
  34. Now we'll convert the cube root of 16 to a mixed radical with the perfect cube method.
  35. Start with the cube root of 16.
  36. Find two numbers that multiply to 16, one of which must be a perfect cube.
  37. Split the radical.
  38. The cube root of 8 is 2. The cube root of 2 is an irrational number, so leave it as a radical. The answer is 2 times the cube root of 2.

 

Example 2: Converting Entire Radicals to Mixed Radicals.

 

  1. In this example, we'll convert entire radicals to mixed radicals. In part a, we'll convert root(24) to a mixed radical.
  2. First we'll use the prime factorization method.
  3. Start with root(24).
  4. Find any two numbers that multiply to 24. We'll use 2 and 12 in this example.
  5. 12 breaks down to 2 and 6.
  6. 6 breaks down to 2 and 3. We now have the prime factorization of 24.
  7. There is a pair of 2's.
  8. The pair of 2's collapse to a single 2 on the outside. Inside the radical, the 2 and 3 multiply to become 6. The answer is 2root(6).
  9. Now we'll convert root(24) to a mixed radical using the perfect square method.
  10. Start with root(24).
  11. Find two numbers that multiply to 24, one of which is a perfect square. Let's use 4 times 6.
  12. Split the radical.
  13. The square root of 4 is 2. The square root of 6 is irrational, so we'll leave this as a radical. The answer is 2root(6).
  14. Now we'll convert root(72) to a mixed radical.
  15. We'll start with the prime factorization method.
  16. Rewrite root(72).
  17. Find any two numbers that multiply to 72. We'll use 9 × 8.
  18. 9 breaks down to 3 × 3, and 8 breaks down to 2 × 2 × 2. We now have the prime factorization of 72.
  19. There is one pair of 3's and one pair of 2's.
  20. The pair of threes collapse to a single 3 outside the radical, and the pair of 2's collapses to a single 2 outside the radical. These values are connected through multiplication. Inside the radical, we have a 2 remaining.
  21. Simplify to get 6root(2). This is the answer.
  22. Now we'll convert root(72) to a mixed radical using the perfect square method.
  23. Start with root(72).
  24. Write 72 as a product of two numbers, one of which must be a perfect square. Let's use 9 × 8.
  25. Split the radical.
  26. This gives us 3root(8). Now be careful! We are not done simplifying since root(8) can be taken down further.
  27. Write 8 as a product of two numbers, one of which must be a perfect square. We'll use 4 × 2.
  28. Split the radical.
  29. This gives us 2root(2).
  30. Simplify to get 6root(2).
  31. In part c, we'll simplify root(49).
  32. 49 is a perfect square, so root(49) is just 7.
  33. Even though we already have the answer, let's see how this could have been done using the techniques we've been using in this lesson.
  34. Start with the prime factorization method.
  35. Rewrite root(49).
  36. 49 breaks down to 7 × 7.
  37. We have a pair of 7's.
  38. If everything is pulled out from underneath the radical, use 1 as a placeholder.
  39. This gives us 7, which is the answer.
  40. Now we'll try this using the perfect square method.
  41. Rewrite root(49).
  42. 49 is 7 × 7.
  43. Split the radical.
  44. When multiplying identical roots, the radical symbol disappears. For example, root(3) times root(3) is just 3. Root 7 times root 7 is just 7, and the cube root of 4 times the cube root of 4 is just 4.
  45. In part d, we'll convert the cube root of 81 to a mixed radical.
  46. Start with the prime factorization method.
  47. Rewrite the cube root of 81.
  48. Find two numbers that multiply to 81. We'll use 9 × 9.
  49. Each 9 breaks down to a pair of 3's. We now have the prime factorization of 81.
  50. Since the index is 3, we need three identical numbers to make a set.
  51. The three 3's collapse to a single three on the outside of the radical, leaving a lone 3 within the radical. The answer is 3 times the cube root of 3.
  52. Now we'll convert the cube root of 81 to a mixed radical using the perfect cube method.
  53. Rewrite the cube root of 81.
  54. Find two numbers that multiply to 81, one of which must be a perfect cube. We'll use 27 times 3.
  55. Split the radical.
  56. The cube root of 27 is 3. The cube root of 3 is an irrational number, so we'll leave this as a radical. The answer is 3 times the cube root of 3.
  57. In part e, we'll find the cube root of 64.
  58. 64 is a perfect cube, so the cube root of 64 is 4.
  59. Even though we have the answer, let's try this using the techniques from this lesson.
  60. Start with the prime factorization method.
  61. Rewrite the cube root of 64.
  62. Find two numbers that multiply to 64.We'll use 8 × 8.
  63. Each 8 becomes 2 × 2 × 2. We now have the prime factorization of 64.
  64. Since the index is 3, we need three in a set. Circle each triplet of 2's.
  65. Pull out the 2's.
  66. Simplify to get our answer, 4.
  67. Now we'll use the perfect cube method.
  68. Rewrite the cube root of 64.
  69. Find two numbers that multiply to 64, one of which must be a perfect cube. Use 8 times 8.
  70. Split the radical.
  71. The cube root of 8 is 2, so we get 2 × 2.
  72. The answer is 4.
  73. In part f, we'll convert the fourth root of 48 to a mixed radical.
  74. We'll start with the prime factorization method.
  75. Rewrite the fourth root of 48.
  76. 48 is the same as 8 × 6.
  77. 8 becomes 2 × 2 × 2 and 6 becomes 2 × 3. We now have the prime factorization of 48.
  78. Since the index is 4, it takes 4 identical numbers to make a set. Circle the set of 4 two's.
  79. The set of 2's collapses to a single two outside the radical, leaving a 3 within the radical. The answer is 2 times the fourth root of 3.
  80. Now we'll convert the fourth root of 48 to a mixed radical using the perfect fourth method.
  81. Rewrite the fourth root of 48.
  82. Find two numbers that multiply to 48, one of which is a perfect fourth. We'll use 16 × 3.
  83. Split the radical.
  84. The fourth root of 16 is 2. The fourth root of 3 is an irrational number, so we'll leave this as a radical. The answer is 2 times the fourth root of 3.

 

Example 3: Converting Mixed Radicals to Entire Radicals.

 

  1. In this example, we will convert each of the following mixed radicals to entire radicals. In part a, we'll convert 3root(3) to an entire radical.
  2. The first method we'll use is called the reverse factorization method.
  3. Rewrite 3root(3).
  4. The 3 outside the radical can be expanded inside the radical as a pair of 3's.
  5. Multiply everything under the radical symbol together to get 27. The entire radical is root(27).
  6. Now we'll convert 3root(3) to an entire radical using the perfect square method.
  7. Rewrite 3root(3).
  8. 3 can be written as root(9).
  9. Multiply the radicals together to get root(27).
  10. In part b, we'll convert 6root(2) to an entire radical.
  11. Start with the reverse factorization method.
  12. Rewrite 6root(2).
  13. The 6 on the outside of the radical can be expanded inside the radical as a pair of 6's.
  14. Multiply everything underneath the radical symbol together to get root(72).
  15. Now we'll convert 6root(2) to an entire radical using the perfect square method.
  16. Rewrite 6root(2).
  17. 6 is the same as root(36).
  18. Multiply the radicals together to get root(72).
  19. In part c, we'll convert 2 times the cube root of 5 to an entire radical.
  20. Start with the reverse factorization method.
  21. Rewrite 2 times the cube root of 5.
  22. The 2 can be expanded inside the radical as a triplet.
  23. Multiply everything underneath the radical to get 40. The entire radical is the cube root of 40.
  24. Now we'll use the perfect cube method.
  25. Rewrite 2 times the cube root of 5.
  26. 2 can be written as the cube root of 8.
  27. Multiply the radicals together to get the cube root of 40.

 

Example 4: Converting Mixed Radicals to Entire Radicals.

 

  1. In this example, convert each mixed radical to an entire radical using the method of your choice. In part a, we'll convert 4root(2) to an entire radical.
  2. Start with the reverse factorization method.
  3. Rewrite 4root(2).
  4. The coefficient of 4 can be expanded inside the radical as a pair of 4's.
  5. Multiply everything underneath the radical symbol to get 32. The entire radical is root(32).
  6. Now we'll convert 4root2 to an entire radical using the perfect square method.
  7. Rewrite 4root(2).
  8. 4 can be written as root16.
  9. Multiply the radicals to get root(32).
  10. In part b, we'll convert 5root(3) to an entire radical.
  11. Start with the reverse factorization method.
  12. Rewrite 5root(3).
  13. The 5 can be expanded inside the radical as a pair of 5's.
  14. Multiply everything underneath the radical to get root(75).
  15. Now we'll do this using the perfect square method.
  16. Rewrite 5root(3).
  17. 5 can be written as root(25).
  18. Multiply the radicals together to get root(75).
  19. In part c, we'll convert 3 times the cube root of 3 to an entire radical.
  20. Start with the reverse factorization method.
  21. Rewrite 3 times the cube root of 3.
  22. The 3 on the outside of the radical can be expanded inside the radical as a triplet.
  23. Multiply all the numbers together to get 81. The entire radical is the cube root of 81.
  24. Now we'll use the perfect cube method.
  25. Rewrite 3 times the cube root of 3.
  26. The coefficient of 3 can be written as the cube root of 27.
  27. Multiply the radicals to get the cube root of 81. This is the entire radical.
  28. In part d, we'll convert 2 times the fourth root of 3 to an entire radical.
  29. Start with the reverse factorization method.
  30. Rewrite 2 times the fourth root of 3.
  31. The coefficient of 2 can be expanded inside the radical asa set of four 2's.
  32. Multiply all the values underneath the radical to get 48. The entire radical is the fourth root of 48.
  33. Now we'll use the perfect fourth method.
  34. Rewrite 2 times the fourth root of 3.
  35. The coefficient 2 can be written as the fourth root of 16.
  36. Multiply the radicals to get the fourth root of 48. This is the entire radical.

 

Example 5: Estimating Radicals.

 

  1. In this example, we will estimate each radical and order them on a number line.
  2. We'll start by estimating root(42).
  3. 42 is approximately halfway between the perfect squares 36 and 49. Since the square root of 36 is 6, and the square root of 49 is 7, the square root of 42 is about 6.5.
  4. Next we'll estimate root(20).
  5. 20 is approximately halfway between the perfect squares 16 and 25. Since the square root of 16 is 4, and the square root of 25 is 5, the square root of 20 is about 4.5.
  6. Next we'll estimate root(8).
  7. 8 is 4/5 of the way between the perfect squares 4 and 9. Since the square root of 4 is 2, and the square root of 9 is 3, the square root of 8 is about 2.8.
  8. Now we'll estimate root(14).
  9. 14 is approximately 3/4 of the way between the perfect squares 9 and 16. Since the square root of 9 is 3, and the square root of 16 is 4, the square root of 14 is about 3.75.
  10. Now we'll order the radicals on a number line.
  11. From left to right, we have root(8), root(14), root(20), and root(42).
  12. Now we'll move on to part b.
  13. We'll start by estimating the cube root of 92.
  14. 92 is approximately halfway between the perfect cubes 64 and 125. Since the cube root of 64 is 4, and the cube root of 125 is 5, the cube root of 92 is about 4.5.
  15. Next we'll estimate the cube root of 169.
  16. 169 is approximately halfway between the perfect cubes 125 and 216. Since the cube root of 125 is 5, and the cube root of 216 is 6, the cube root of 169 is about 5.5.
  17. Next we'll estimate the cube root of 54.
  18. 54 is approximately 3/4 of the way between the perfect cubes 27 and 64. Since the cube root of 27 is 3, and the cube root of 64 is 4, the cube root of 54 is about 3.75.
  19. Next we'll estimate the cube root of 35.
  20. 35 is approximately 1/4 of the way between the perfect cubes 27 and 64. Since the cube root of 27 is 3, and the cube root of 64 is 4, the cube root of 35 is about 3.25.
  21. Now we'll order the radicals on a number line.
  22. From left to right, we have the cube root of (35),the cube root of (54),the cube root of (92), and the cube root of (169).

 

Example 6: Simplifying Radicals.

 

  1. In this example, we will simplify the following expressions without using a calculator. In part a, we'll simplify 2root(12)/4.
  2. First simplify root(12).
  3. Rewrite root(12).
  4. We'll convert root(12) to a mixed radical using prime factorization. Rewrite 12 as 4 times 3.
  5. 4 can be written as 2 × 2.
  6. There is a pair of 2's. The pair collapses to a single 2 outside the radical.
  7. Now replace root(12) in the original question with 2root(3).
  8. Rewrite the original expression.
  9. Replace root(12) with 2root(3).
  10. Simplify the numerator to get 4root(3).
  11. The 4's cancel out.
  12. The answer is root(3).
  13. In part b, we'll simplify 3 times the cube root of 27 over 36.
  14. First evaluate the cube root of 27.
  15. The cube root of 27 is 3.
  16. Now replace the cube root of 27 in the original question with 3.
  17. Rewrite the original expression.
  18. replace the cube root of 27 in the original question with 3.
  19. Simplify to get 9/36.
  20. And finally, reduce the fraction to 1/4.
  21. In part c, we'll simplify 3/4 times root(32).
  22. Begin by converting root(32)to a mixed radical.
  23. Rewrite root(32).
  24. Convert root(32) to a mixed radical with prime factorization. 32 is the same as 4 times 8.
  25. 4 becomes 2 × 2, and 8 becomes 2 × 2 × 2.
  26. There are two pairs of 2's.
  27. Each pair collapses to a single 2 outside the radical.
  28. This simplifies to 4root(2).
  29. Now replace root(32) with 4root(2).
  30. Rewrite the original expression.
  31. Replace root(32) with 4root(2).
  32. 4root(2) can be written in the numerator of the fraction.
  33. The 4's cancel.
  34. We are left with 3root(2) This is the answer.
  35. In part d, we'll simplify root(49/81).
  36. Rewrite root(49/81).
  37. Both the numerator and denominator are perfect squares, giving us 7/9 when we take the square root.
  38. In part e, we'll evaluate 3 times the cube root of 72 over the square root of 64.
  39. First convert the cube root of 72 to a mixed radical.
  40. We'll convert the cube root of 72 to a mixed radical using prime factorization.
  41. 72 is the same as 9 × 8.
  42. 9 is the same as 3 × 3, and 8 is the same as 2 × 2 × 2.
  43. There is a triplet of 2's, so they collapse to a single 2 outside the radical. The 3's remaining within the radical multiply to 9.
  44. Now replace the cube root of 72 with 2 times the cube root of 9.
  45. Rewrite the original expression.
  46. Replace the cube root of 72 with 2 times the cube root of 9.
  47. Simplify the numerator.
  48. 6/8 reduces to 3/4. The final answer is 3 times the cube root of 9 over 4.

 

Example 7: Writing Exponents as Radicals.

 

  1. In this example, we will write each power as a radical.
  2. Before we begin, let's take a moment to learn how to write a power as a radical.
  3. This is a power with a fractional exponent. m is the numerator, and n is the denominator.
  4. We can write this as a radical. m becomes the exponent of the radicand, and n becomes the index.
  5. Alternatively, we can write the radical in brackets and place the exponent m outside.
  6. Now we'll convert 3^(1/2) to a radical.
  7. The denominator of the exponent becomes the index of the radical, and the numerator of the exponent goes to the radicand.
  8. The answer is the square root of 3.
  9. In part b, we'll convert (-4)^(1/3) to a radical.
  10. The denominator of the exponent becomes the index of the radical, and the numerator of the exponent goes to the radicand.
  11. The answer is the cube root of -4.
  12. In part c, we'll convert 2^(4/3)to a radical.
  13. We can write this as either the cube root of 24 OR the cube root of 2 in brackets, raised to the power of 4.
  14. In part d, we'll convert (-7)^(2/5) to a radical.
  15. We can write this as the fifth root of -7 squared, OR the fifth root of -7, in brackets, raised to the power of 2.
  16. In part e, we'll convert (2/3)^(3/2) to a radical.
  17. We can write this as the square root of (2/3) cubed, OR, the square root of (2/3), in brackets, raised to the power of 3.
  18. In part f, we'll convert 16^0.25 to a radical.
  19. We can write 0.25 as 1/4.
  20. This gives us the fourth root of 16.

 

Example 8: Writing Radicals as Exponents.

 

  1. In this example, we will write each radical as a power. In part a, we will convert root(5) to a power.
  2. The index of the radical is 2, so that becomes the denominator of the exponent. The radicand has an exponent of 1, so that becomes the numerator.
  3. In part b, we will convert the fourth root of 9 to a power.
  4. The index of the radical is 4, so that becomes the denominator of the exponent. The radicand has an exponent of 1, so that becomes the numerator.
  5. In part c, we'll convert the cube root of 22 to a power.
  6. The index of the radical is 3, so that becomes the denominator. The radicand has an exponent of 2, so that becomes the numerator.
  7. In part d, we'll convert the fifth root of -3, in brackets, to the fourth power.
  8. The index of the radical is 5, so that becomes the denominator. The radical has an exponent of 4, so that becomes the numerator. In our power, we must put brackets around -3. Whenever the base is negative, it must be put in brackets.
  9. In part e, we have the cube root of (5/7), in brackets, raised to the power of 2.
  10. The index of the radical is 3, so that becomes the denominator. The radical has an exponent of 2, so that becomes the numerator.
  11. In part f, we'll convert the square root of (3/4) squared to a radical.
  12. The index of the radical is 2, so that becomes the denominator.
  13. The radicand has an exponent of 2, so that becomes the numerator.
  14. The exponent divides to 1.
  15. The answer is 3/4.