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Mathematics LibreTexts

5.1: Roots and Radicals

  • Anonymous
  • LibreTexts

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Learning Objectives

  • Identify and evaluate square and cube roots.
  • Determine the domain of functions involving square and cube roots.
  • Evaluate nth roots.
  • Simplify radicals using the product and quotient rules for radicals.

Square and Cube Roots

Recall that a square root1 of a number is a number that when multiplied by itself yields the original number. For example, 5 is a square root of 25, because 52=25. Since (5)2=25, we can say that 5 is a square root of 25 as well. Every positive real number has two square roots, one positive and one negative. For this reason, we use the radical sign to denote the principal (nonnegative) square root2 and a negative sign in front of the radical to denote the negative square root.

25=5Positivesquarerootof2525=5Negativesquarerootof25

Zero is the only real number with one square root.

0=0 because 02=0

Example 5.1.1:

Evaluate.

  1. 121
  2. 81

Solution

  1. 121=112=11
  2. 81=92=9

If the radicand3, the number inside the radical sign, can be factored as the square of another number, then the square root of the number is apparent. In this case, we have the following property:

a2=a if a0

Or more generally,

a2=|a| if aR

The absolute value is important because a may be a negative number and the radical sign denotes the principal square root. For example,

(8)2=|8|=8

Make use of the absolute value to ensure a positive result.

Example 5.1.2:

Simplify: (x2)2.

Solution

Here the variable expression x2 could be negative, zero, or positive. Since the sign depends on the unknown quantity x, we must ensure that we obtain the principal square root by making use of the absolute value.

(x2)2=|x2|

Answer:

|x2|

The importance of the use of the absolute value in the previous example is apparent when we evaluate using values that make the radicand negative. For example, when x=1,

(x2)2=|x2|=|12|=|1|=1

Next, consider the square root of a negative number. To determine the square root of 25, you must find a number that when squared results in 25:

25=? or (?)2=25

However, any real number squared always results in a positive number. The square root of a negative number is currently left undefined. For now, we will state that 25 is not a real number. Therefore, the square root function4 given by f(x)=x is not defined to be a real number if the x-values are negative. The smallest value in the domain is zero. For example,f(0)=0=0 and f(4)=4=2. Recall the graph of the square root function.

5a5bec559b23c37c3dc80885e5f0026c.png
Figure 5.1.1

The domain and range both consist of real numbers greater than or equal to zero: [0,). To determine the domain of a function involving a square root we look at the radicand and find the values that produce nonnegative results.

Example 5.1.3:

Determine the domain of the function defined by f(x)=2x+3.

Solution

Here the radicand is 2x+3. This expression must be zero or positive. In other words,

2x+30

Solve for x.

2x+302x3x32

Answer:

Domain: [32,)

A cube root5 of a number is a number that when multiplied by itself three times yields the original number. Furthermore, we denote a cube root using the symbol 3, where 3 is called the index6. For example,

364=4, because 43=64

The product of three equal factors will be positive if the factor is positive and negative if the factor is negative. For this reason, any real number will have only one real cube root. Hence the technicalities associated with the principal root do not apply. For example,

364=4, because (4)3=64

In general, given any real number a, we have the following property:

3a3=a if aR

When simplifying cube roots, look for factors that are perfect cubes.

Example 5.1.4:

Evaluate.

  1. 38
  2. 30
  3. 3127
  4. 31
  5. 3125

Solution

  1. 38=323=2
  2. 30=303=0
  3. 3127=3(13)3=13
  4. 31=3(1)3=1
  5. 3125=3(5)3=5

It may be the case that the radicand is not a perfect square or cube. If an integer is not a perfect power of the index, then its root will be irrational. For example, 32 is an irrational number that can be approximated on most calculators using the root button x.Depending on the calculator, we typically type in the index prior to pushing the button and then the radicand as follows:

3xy2=

Therefore, we have

321.260, because 1.26032

Since cube roots can be negative, zero, or positive we do not make use of any absolute values.

Example 5.1.5:

Simplify: 3(y7)3.

Solution

The cube root of a quantity cubed is that quantity.

3(y7)3=y7

Answer:

y7

Exercise 5.1.1

Evaluate: 31000.

Answer

=10

www.youtube.com/v/B06NIs-3gig

Next, consider the cube root function7:

f(x)=3xCuberootfunction.

Since the cube root could be either negative or positive, we conclude that the domain consists of all real numbers. Sketch the graph by plotting points. Choose some positive and negative values for x, as well as zero, and then calculate the corresponding y-values.

x f(x) f(x)=3x OrderedPairs
8 2 f(8)=38=2 (8,2)
1 1 f(1)=31=1 (1,1)
0 0 f(0)=30=0 (0,0)
1 1 f(1)=31=1 (1,1)
8 2 f(8)=38=2 (8,2)
Table 5.1.1

Plot the points and sketch the graph of the cube root function.

507f092c47524abca6b1b6d100809a5e.png
Figure 5.1.2

The graph passes the vertical line test and is indeed a function. In addition, the range consists of all real numbers.

Example 5.1.6:

Given g(x)=3x+1+2, find g(9),g(2),g(1), and g(0). Sketch the graph of g.

Solution

Replace x with the given values.

x g(x) g(x)=3x+1+2 OrderedPairs
9 0 g(9)=39+1+2=38+2=2+2=0 (9,0)
2 1 g(2)=32+1+2=31+2=1+2=1 (2,1)
1 2 g(1)=31+1+2=30+2=0+2=2 (1,2)
0 3 g(0)=30+1+2=31+2=1+2=3 (0,3)
Table 5.1.2

We can also sketch the graph using the following translations:

y=3xBasiccuberootfunctiony=3x+1Horizontalshiftleft1unity=3x+1+2Verticalshiftup2units

Answer:

c7808cb98ec1b47125d20c368e032309.png
Figure 5.1.3

nth Roots

For any integer n2, we define an nth root8 of a positive real number as a number that when raised to the nth power yields the original number. Given any nonnegative real number a, we have the following property:

nan=a, if a0

Here n is called the index and an is called the radicand. Furthermore, we can refer to the entire expression nA as a radical9. When the index is an integer greater than or equal to 4, we say “fourth root,” “fifth root,” and so on. The nth root of any number is apparent if we can write the radicand with an exponent equal to the index.

Example 5.1.7:

Simplify:

  1. 481
  2. 532
  3. 71
  4. 4116

Solution

  1. 481=434=3
  2. 532=525=2
  3. 71=717=1
  4. 4116=4(12)4=12

Note

If the index is n=2, then the radical indicates a square root and it is customary to write the radical without the index; 2a=a.

We have already taken care to define the principal square root of a real number. At this point, we extend this idea to nth roots when n is even. For example, 3 is a fourth root of 81, because 34=81. And since (3)4=81, we can say that 3 is a fourth root of 81 as well. Hence we use the radical sign n to denote the principal (nonnegative) nth root10 when n is even. In this case, for any real number a, we use the following property:

nan=|a|Whenniseven

For example,

481=434=|3|=3481=4(3)4=|3|=3

The negative nth root, when n is even, will be denoted using a negative sign in front of the radical n.

481=434=3

We have seen that the square root of a negative number is not real because any real number that is squared will result in a positive number. In fact, a similar problem arises for any even index:

481=? or (?)4=81

We can see that a fourth root of 81 is not a real number because the fourth power of any real number is always positive.

4481664}Theseradicalsarenotrealnumbers.

You are encouraged to try all of these on a calculator. What does it say?

Example 5.1.8:

Simplify.

  1. 4(10)4
  2. 4104
  3. 6(2y+1)6

Solution

Since the indices are even, use absolute values to ensure nonnegative results.

  1. 4(10)4=|10|=10
  2. 4104=410,000 is not a real number.
  3. 6(2y+1)6=|2y+1|

When the index n is odd, the same problems do not occur. The product of an odd number of positive factors is positive and the product of an odd number of negative factors is negative. Hence when the index n is odd, there is only one real nth root for any real number a. And we have the following property:

nan=aWhennisodd

Example 5.1.9:

Simplify.

  1. 5(10)5
  2. 532
  3. 7(2y+1)7

Solution

Since the indices are odd, the absolute value is not used.

  1. 5(10)5=10
  2. 532=5(2)5=2
  3. 7(2y+1)7=2y+1

In summary, for any real number a we have,

nan=|a|Whennisevennan=aWhennisodd

When n is odd, the nth root is positive or negative depending on the sign of the radicand.

327=333=3327=3(3)3=3

When n is even, the nth root is positive or not real depending on the sign of the radicand.

416=424=2416=4(2)4=|2|=2416Notarealnumber

Exercise 5.1.2

Simplify: 8532.

Answer

16

www.youtube.com/v/Ik1xXgq18f0

Simplifying Radicals

It will not always be the case that the radicand is a perfect power of the given index. If it is not, then we use the product rule for radicals11 and the quotient rule for radicals12 to simplify them. Given real numbers nA and nB,

Product Rule for Radicals: nAB=nAnB
Quotient Rule for Radicals: nAB=nAnB
Table 5.1.3

A radical is simplified13 if it does not contain any factors that can be written as perfect powers of the index.

Example 5.1.10:

Simplify: 150.

Solution

Here 150 can be written as 2352.

150=2352Applytheproductruleforradicals.=2352Simplify.=65=56

We can verify our answer on a calculator:

15012.25 and 5612.25

Also, it is worth noting that

12.252150

Answer:

56

Note

56 is the exact answer and 12.25 is an approximate answer. We present exact answers unless told otherwise.

Example 5.1.11:

Simplify: 3160.

Solution

Use the prime factorization of 160 to find the largest perfect cube factor:

160=255=23225

Replace the radicand with this factorization and then apply the product rule for radicals.

3160=323225Applytheproductruleforradicals.=3233225Simplify.=2320

We can verify our answer on a calculator.

31605.43 and 23205.43

Answer:

2320

Example 5.1.12:

Simplify: 5320.

Solution

Here we note that the index is odd and the radicand is negative; hence the result will be negative. We can factor the radicand as follows:

320=13210=(1)5(2)510

Then simplify:

5320=5(1)5(2)510Applytheproductruleforradicals.=5(1)55(2)5510Simplify.=12510=2510

Answer:

2510

Example 5.1.13:

Simplify: 3864.

Solution

In this case, consider the equivalent fraction with 8=(2)3 in the numerator and 64=43 in the denominator and then simplify.

3864=3864Applythequotientruleforradicals.=3(2)3343Simplify.=24=12

Answer:

12

Exercise 5.1.3

Simplify: 48081

Answer

2453

www.youtube.com/v/8CwbDBFO2FQ

Key Takeaways

  • To simplify a square root, look for the largest perfect square factor of the radicand and then apply the product or quotient rule for radicals.
  • To simplify a cube root, look for the largest perfect cube factor of the radicand and then apply the product or quotient rule for radicals.
  • When working with nth roots, n determines the definition that applies. We use nan=a1 when n is odd and nan=|a| when n is even.
  • To simplify nth roots, look for the factors that have a power that is equal to the index n and then apply the product or quotient rule for radicals. Typically, the process is streamlined if you work with the prime factorization of the radicand.

Exercise 5.1.4

Simplify.

  1. 36
  2. 100
  3. 49
  4. 164
  5. 16
  6. 1
  7. (5)2
  8. (1)2
  9. 4
  10. 52
  11. (3)2
  12. (4)2
  13. x2
  14. (x)2
  15. (x5)2
  16. (2x1)2
  17. 364
  18. 3216
  19. 3216
  20. 364
  21. 38
  22. 31
  23. 3(2)3
  24. 3(7)3
  25. 318
  26. 3827
  27. 3(y)3
  28. 3y3
  29. 3(y8)3
  30. 3(2x3)3
Answer

1. 6

3. 23

5. 4

7. 5

9. Not a real number

11. 3

13. |x|

15. |x5|

17. 4

19. 6

21. 2

23. 2

25. 12

27. y

29. y8

Exercise 5.1.5

Determine the domain of the given function.

  1. g(x)=x+5
  2. g(x)=x2
  3. f(x)=5x+1
  4. f(x)=3x+4
  5. g(x)=x+1
  6. g(x)=x3
  7. h(x)=5x
  8. h(x)=23x
  9. g(x)=3x+4
  10. g(x)=3x3
Answer

1. [5,)

3. [15,)

5. (,1]

7. (,5]

9. (,)

Exercise 5.1.6

Evaluate given the function definition.

  1. Given f(x)=x1, find f(1),f(2), and f(5)
  2. Given f(x)=x+5, find f(5),f(1), and f(20)
  3. Given f(x)=x+3, find f(0),f(1), and f(16)
  4. Given f(x)=x5, find f(0),f(1), and f(25)
  5. Given g(x)=3x, find g(1),g(0), and g(1)
  6. Given g(x)=3x2 find g(1),g(0), and g(8)
  7. Given g(x)=3x+7, find g(15),g(7), and g(20)
  8. Given g(x)=3x1+2, find g(0),g(2), and g(9)
Answer

1. f(1)=0;f(2)=1;f(5)=2

3. f(0)=3;f(1)=4;f(16)=7

5. g(1)=1;g(0)=0;g(1)=1

7. g(15)=2;g(7)=0;g(20)=3

Exercise 5.1.7

Sketch the graph of the given function and give its domain and range.

  1. f(x)=x+9
  2. f(x)=x3
  3. f(x)=x1+2
  4. f(x)=x+1+3
  5. g(x)=3x1
  6. g(x)=3x+1
  7. g(x)=3x4
  8. g(x)=3x+5
  9. g(x)=3x+21
  10. g(x)=3x2+3
  11. f(x)=3x
  12. f(x)=3x1
Answer

1. Domain: [9,); range: [0,)

d41e03bbc0db0d0bbb412151227f80cf.png
Figure 5.1.4

3. Domain: [1,); range: [2,)

326c85b7ef2b6468784f06604da2e200.png
Figure 5.1.5

5. Domain: R; range; R

9da27c50cbdbe8f826496fa49c70e01a.png
Figure 5.1.6

7. Domain: R; range; R

e17bdac87837ac3587ddf019eb7226ba.png
Figure 5.1.7

9. Domain: R; range; R

b84dad2e3ddd99f6bf8eca4ab9612020.png
Figure 5.1.8

11. Domain: R; range; R

b7cb2d25f3f876f0c587f47f97e8fa68.png
Figure 5.1.9

Exercise 5.1.8

Simplify.

  1. 464
  2. 416
  3. 4625
  4. 41
  5. 4256
  6. 410,000
  7. 5243
  8. 5100,000
  9. 5132
  10. 51243
  11. 416
  12. 61
  13. 532
  14. 51
  15. 1
  16. 416
  17. 6327
  18. 538
  19. 231,000
  20. 75243
  21. 6416
  22. 12664
  23. 32516
  24. 6169
  25. 5327125
  26. 753275
  27. 53827
  28. 8462516
  29. 25100,000
  30. 27128
Answer

1. 4

3. 5

5. 4

7. 3

9. 12

11. 2

13. 2

15. Not a real number

17. 18

19. 20

21. Not a real number

23. 154

25. 3

27. 103

29. 20

Exercise 5.1.9

Simplify.

  1. 96
  2. 500
  3. 480
  4. 450
  5. 320
  6. 216
  7. 5112
  8. 10135
  9. 2240
  10. 3162
  11. 15049
  12. 2009
  13. 675121
  14. 19281
  15. 354
  16. 324
  17. 348
  18. 381
  19. 340
  20. 3120
  21. 3162
  22. 3500
  23. 354125
  24. 340343
  25. 5348
  26. 23108
  27. 8496
  28. 74162
  29. 5160
  30. 5486
  31. 5224243
  32. 5532
  33. 5132
  34. 6164
Answer

1. 46

3. 430

5. 85

7. 207

9. 815

11. 567

13. 15311

15. 332

17. 236

19. 235

21. 336

23. 3325

25. 1036

27. 1646

29. 255

31. 2573

33. 12

Exercise 5.1.10

Simplify. Give the exact answer and the approximate answer rounded to the nearest hundredth.

  1. 60
  2. 600
  3. 9649
  4. 19225
  5. 3240
  6. 3320
  7. 3288125
  8. 36258
  9. 4486
  10. 5288
Answer

1. 215;7.75

3. 467;1.40

5. 2330;6.21

7. 23365;1.32

9. 346;4.70

Exercise 5.1.11

Rewrite the following as a radical expression with coeffecient 1.

  1. 215
  2. 37
  3. 510
  4. 103
  5. 237
  6. 336
  7. 245
  8. 342
  9. Each side of a square has a length that is equal to the square root of the square’s area. If the area of a square is 72 square units, find the length of each of its sides.
  10. Each edge of a cube has a length that is equal to the cube root of the cube’s volume. If the volume of a cube is 375 cubic units, find the length of each of its edges.
  11. The current I measured in amperes is given by the formula I=PR where P is the power usage measured in watts and R is the resistance measured in ohms. If a 100 watt light bulb has 160 ohms of resistance, find the current needed. (Round to the nearest hundredth of an ampere.)
  12. The time in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet the object has fallen. How long will it take an object to fall to the ground from the top of an 8-foot stepladder? (Round to the nearest tenth of a second.)
Answer

1. 60

3. 250

5. 356

7. 480

9. 62 units

11. 0.79 ampere

Exercise 5.1.12

  1. Explain why there are two real square roots for any positive real number and one real cube root for any real number.
  2. What is the square root of 1 and what is the cube root of 1? Explain why.
  3. Explain why 1 is not a real number and why 31 is a real number.
  4. Research and discuss the methods used for calculating square roots before the common use of electronic calculators.
Answer

1. Answer may vary

3. Answer may vary

Footnotes

1A number that when multiplied by itself yields the original number.

2The positive square root of a positive real number, denoted with the symbol .

3The expression A within a radical sign, nA.

4The function defined by f(x)=x.

5A number that when used as a factor with itself three times yields the original number, denoted with the symbol 3.

6The positive integer n in the notation n that is used to indicate an nth root.

7The function defined by f(x)=3x.

8A number that when raised to the nth power (n2) yields the original number.

9Used when referring to an expression of the form nA.

10The positive nth root when n is even.

11Given real numbers nA and nB,nAB=nAnB.

12Given real numbers nA and nB,nAB=nAnB where B0.

13A radical where the radicand does not consist of any factors that can be written as perfect powers of the index.


This page titled 5.1: Roots and Radicals is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.

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