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This is “Radical Expressions and Equations”, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
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i
Chapter 8
Radical Expressions and Equations
Example 1: Find the square root.
a. 36
⎯ ⎯⎯⎯ √
b. 144
⎯⎯⎯⎯⎯ ⎯ √
c. 0.
⎯⎯⎯⎯⎯⎯ ⎯ √
d. (^19)
⎯ ⎯⎯ √
Solution:
a. 36
⎯ ⎯⎯⎯ √ = 62
⎯ ⎯⎯⎯ √ (^) = 6
b. 144
⎯⎯⎯⎯⎯ ⎯ √ = 122
⎯⎯⎯⎯⎯ ⎯ √ (^) = 12
c. 0.
⎯⎯⎯⎯⎯⎯ ⎯ √ = (0.2)^2
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √ (^) = 0.
d. (^19)
⎯ ⎯⎯ √ =^ (^
1 3 )
⎯⎯⎯⎯⎯⎯⎯ 2 ⎯ √ =^
1 3
Example 2: Find the negative square root.
a. − 4
⎯⎯ √
b. − 1
⎯⎯ √
Solution:
a. − 4
⎯⎯ √ = − 22
⎯ ⎯⎯⎯ √ (^) = −
b. − 1
⎯⎯ √ = − 12
⎯ ⎯⎯⎯ √ (^) = −
The radicand may not always be a perfect square. If a positive integer is not a perfect square, then its square root will be irrational. For example, 2
⎯⎯ √ is an irrational number and can be approximated on most calculators using the square root button.
Next, consider the square root of a negative number. To determine the square root of −9, you must find a number that when squared results in −9:
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 −
⎯ ⎯⎯⎯⎯ √ is not a real a number.
Cube Roots
The cube root^4 of a number is that 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 index^5. For example,
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
- The number that, when used as a factor with itself three times, yields the original number; it is
denoted with the symbol √^3.
- The positive integer n in the
notation √ n^ that is used to
indicate an n th root.
d. (^18)
⎯ ⎯⎯ √
(^3) = (
1 2 )
⎯⎯⎯⎯⎯⎯⎯ 3 ⎯ √
(^3) = 1 2
Example 4: Find the cube root.
a. −
⎯ ⎯⎯⎯⎯ √^3
b. −
⎯ ⎯⎯⎯⎯ √^3
c. − 271
⎯⎯⎯⎯⎯⎯ ⎯ √
3
Solution:
a. −
⎯ ⎯⎯⎯⎯ √^3 = (−2)^3
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √^3 = −
b. −
⎯ ⎯⎯⎯⎯ √^3 = (−1)^3
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √^3 = −
c. − 271
⎯⎯⎯⎯⎯⎯ ⎯ √
(^3) = (−^
1 3 )
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ 3 ⎯ √
(^3) = − 1 3
It may be the case that the radicand is not a perfect cube. If an integer is not a
perfect cube, then its cube root will be irrational. For example, 2
⎯⎯ √^3 is an irrational number which can be approximated on most calculators using the root button. Depending on the calculator, we typically type in the index prior to pushing the button and then the radicand as follows:
Therefore, we have
n th Roots
For any integer n ≥ 2, we define the nth root^6 of a positive real number as that number that when raised to the n th power yields the original number. Given any nonnegative real number a , we have the following property:
Here n is called the index and an^ is called the radicand. Furthermore, we can refer to the entire expression a ⎯⎯ √ n^ as a^ radical^7. When the index is an integer greater than 3, we say “fourth root”, “fifth root”, and so on. The n th root of any number is apparent if we can write the radicand with an exponent equal to the index.
Example 5: Find the n th root.
a. 81
⎯ ⎯⎯⎯ √^4
b. 32
⎯ ⎯⎯⎯ √^5
c. 1
⎯⎯ √^7
d. 161
⎯ ⎯⎯⎯ √
4
Solution:
a. 81
⎯ ⎯⎯⎯ √^4 = 34
⎯ ⎯⎯⎯ √ 4 = 3
b. 32
⎯ ⎯⎯⎯ √^5 =^25
⎯ ⎯⎯⎯ √^5 = 2
c. 1
⎯⎯ √^7 = 17
⎯ ⎯⎯⎯ √^7 = 1
- The number that, when raised to the n th power, yields the original number.
- Used when referring to an
expression of the form a
√ n^.
Here the fourth root of −81 is not a real number because the fourth power of any real number is always positive.
Example 6: Simplify.
a. −
⎯⎯⎯⎯⎯⎯ ⎯ √^4
b. − 16
⎯ ⎯⎯⎯ √^4
Solution:
a. The radicand is negative and the index is even. Therefore, there is no real number that when raised to the fourth power is −16.
b. Here the radicand is positive. Furthermore, 16 = 2^4 , and we can simplify as follows:
When 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 n th root for any real number a. And we have the following property:
Example 7: Find the n th root.
a. −
⎯⎯⎯⎯⎯⎯ ⎯ √^5
b. −
⎯ ⎯⎯⎯⎯ √^7
Solution:
a. −
⎯⎯⎯⎯⎯⎯ ⎯ √^5 = (−2)^5
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √
(^5) = −
b. −
⎯ ⎯⎯⎯⎯ √^7 = (−1)^7
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √^7 = −
Try this! Find the fourth root: 625
⎯⎯⎯⎯⎯ ⎯ √^4.
Answer: 5
Video Solution
(click to see video)
Summary: When n is odd , the n th root is positive or negative depending on the sign of the radicand.
We can verify our answer on a calculator:
Also, it is worth noting that
Answer: 2 3
⎯⎯ √
Example 9: Simplify: 135
⎯⎯⎯⎯⎯ ⎯ √.
Solution: Begin by finding the largest perfect square factor of 135.
Therefore,
Answer: 3 15
⎯ ⎯⎯⎯ √
Example 10: Simplify: 12150
⎯⎯⎯⎯⎯ ⎯ √.
Solution: Begin by finding the prime factorizations of both 50 and 121. This will enable us to easily determine the largest perfect square factors.
Therefore,
Answer:
5 √ 2 11
Example 11: Simplify: 162
⎯⎯⎯⎯⎯ ⎯ √^3.
Solution: Use the prime factorization of 162 to find the largest perfect cube factor:
Then simplify:
Answer: −2 3
⎯⎯ √ 5
Example 13: Simplify: − 648
⎯⎯⎯⎯⎯⎯ ⎯ √
Solution: In this case, consider the equivalent fraction with −8 = (−2)^3 in the numerator and then simplify.
Answer: −1/
Try this! Simplify −
⎯⎯⎯⎯⎯⎯⎯ ⎯ √
Answer: −3 4
⎯⎯ √^3
Video Solution
(click to see video)
K E Y T A K E A W A Y S
- The square root of a number is that number that when multiplied by itself yields the original number. When the radicand a is positive, a^2
⎯ ⎯⎯⎯ √ (^) = a. When the radicand is negative, the result is not a real number.
- The cube root of a number is that number that when used as a factor with itself three times yields the original number. The cube root may be positive or negative depending on the sign of the radicand. Therefore, for any real number a , we have the property a^3
⎯ ⎯⎯⎯ √ 3 = a.
- When working with n th roots, n determines the definition that applies. We use an
⎯ ⎯⎯⎯ √
n = a when n is odd and an
⎯ ⎯⎯⎯ √
n = | a |when n is even. When n is even, the negative n th root is denoted with a negative sign in front of the radical sign.
- To simplify square roots, look for the largest perfect square factor of the radicand and then apply the product or quotient rule for radicals.
- To simplify cube roots, look for the largest perfect cube factor of the radicand and then apply the product or quotient rule for radicals.
- To simplify n th 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.
- −
⎯⎯⎯⎯⎯⎯ ⎯ √
- −
⎯⎯⎯⎯ ⎯ √
- − 36
⎯ ⎯⎯⎯ √
- − 81
⎯ ⎯⎯⎯ √
- − 100
⎯⎯⎯⎯⎯ ⎯ √
- − 1
⎯⎯ √
- 27
⎯ ⎯⎯⎯ √ 3
- 125
⎯⎯⎯⎯⎯ ⎯ √ 3
- 64
⎯ ⎯⎯⎯ √ 3
- 8
⎯⎯ √ 3
⎯ ⎯⎯ √
3
⎯ ⎯⎯⎯⎯ √
3
⎯ ⎯⎯⎯⎯ √
3
⎯⎯⎯⎯⎯ ⎯ √
3
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √ 3
- 1,
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √ 3
- −
⎯⎯⎯⎯ ⎯ √ 3
- −
⎯⎯⎯⎯ ⎯ √ 3
- −
⎯⎯⎯⎯⎯⎯ ⎯ √ 3
- −
⎯⎯⎯⎯⎯⎯ ⎯ √ 3
- − (^18)
⎯⎯⎯⎯⎯ ⎯ √
3
- − (^2764)
⎯⎯⎯⎯⎯⎯⎯ ⎯ √
3
- − 278
⎯⎯⎯⎯⎯⎯⎯ ⎯ √
3
- − 1251
⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √
3
- 81
⎯ ⎯⎯⎯ √ 4
- 625
⎯⎯⎯⎯⎯ ⎯ √ 4
- 16
⎯ ⎯⎯⎯ √ 4
- 10,
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √ 4
- 32
⎯ ⎯⎯⎯ √ 5
- 1
⎯⎯ √ 5
- 243
⎯⎯⎯⎯⎯ ⎯ √ 5
- 100,
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯ √ 5
