We know that 52=25, and 25=5, but what if we want to "undo" 53=125, or 54=625? We can use higher order roots to answer these questions.
While square roots are probably the most common radical, you can also find the third root, the fifth root, the 10th root, or really any other nth root of a number. Just as the square root is a number that, when squared, gives the radicand, the cube root is a number that, when cubed, gives the radicand.
The cube root of a number is written with a small number 3, called the index, just outside and above the radical symbol. It looks like 3. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.
Caution! Be careful to distinguish between 3x, the cube root of x, and 3x, three times the square root of x. They may look similar at first, but they lead you to much different expressions!
Suppose we know that a3=8. We want to find what number raised to the 3rd power is equal to 8. Since 23=8, we say that 2 is the cube root of 8. In the next example, we will evaluate the cube roots of some perfect cubes.
Example
Evaluate the following:
3−8
327
0
Answer:
1. We want to find a number whose cube is −8. We know 2 is the cube root of 8, so maybe we can try −2 which gives −2⋅−2⋅−2=−8, so the cube root of −8 is −2. This is different from square roots because multiplying three negative numbers together results in a negative number.
2. We want to find a number whose cube is 27. 3⋅3⋅3=27, so the cube root of 27 is 3.
3. We want to find a number whose cube is 0. 0⋅0⋅0, no matter how many times you multiply 0 by itself, you will always get 0.
As we saw in the last example, there is one interesting fact about cube roots that is not true of square roots. Negative numbers cannot have real number square roots, but negative numbers can have real number cube roots! What is the cube root of −8? 3−8=−2 because −2⋅−2⋅−2=−8. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider 3(−1)3=−1.
We can also use factoring to simplify cube roots such as 3125. You can read this as “the third root of 125” or “the cube root of 125.” To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. Let’s factor 125 and find that number.
Example
Simplify. 3125
Answer: 125 ends in 5, so you know that 5 is a factor. Expand 125 into 5⋅25.
35⋅25
Factor 25 into 5and 5.
35⋅5⋅5
The factors are 5⋅5⋅5, or 53.
353
Answer
3125=5
The prime factors of 125 are 5⋅5⋅5, which can be rewritten as 53. The cube root of a cubed number is the number itself, so 353=5. You have found the cube root, the three identical factors that when multiplied together give 125. 125 is known as a perfect cube because its cube root is an integer.
In the following video, we show more examples of finding a cube root.
https://youtu.be/9Nh-Ggd2VJo
Here’s an example of how to simplify a radical that is not a perfect cube.
Example
Simplify. 332m5
Answer: Factor 32 into prime factors.
32⋅2⋅2⋅2⋅2⋅m5
Since you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite 2⋅2⋅2 as 23.
323⋅2⋅2⋅m5
Rewrite m5 as m3⋅m2.
323⋅2⋅2⋅m3⋅m2
Rewrite the expression as a product of multiple radicals.
323⋅32⋅2⋅3m3⋅3m2
Simplify and multiply.
2⋅34⋅m⋅3m2
Answer
332m5=2m34m2
In the example below, we use the idea that 3(−1)3=−1 to simplify the radical. You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.
Example
Simplify. 3−27x4y3
Answer: Factor the expression into cubes.
Separate the cubed factors into individual radicals.
You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals 3−81,5−64, and 7−2187, but you cannot simplify the radicals −100,4−16, or 6−2,500.
Let’s look at another example.
Example
Simplify. 3−24a5
Answer: Factor −24 to find perfect cubes. Here, −1 and 8 are the perfect cubes.
3−1⋅8⋅3⋅a5
Factor variables. You are looking for cube exponents, so you factor a5 into a3 and a2.
3(−1)3⋅23⋅3⋅a3⋅a2
Separate the factors into individual radicals.
3(−1)3⋅323⋅3a3⋅33⋅a2
Simplify, using the property 3x3=x.
−1⋅2⋅a⋅33⋅a2
This is the simplest form of this expression; all cubes have been pulled out of the radical expression.
−2a33a2
Answer
3−24a5=−2a33a2
In the next video, we share examples of finding cube roots with negative radicands.
Try It
[ohm_question]196049[/ohm_question]
https://youtu.be/BtJruOpmHCE
In the same way that we learned earlier that we can estimate square roots, we can also estimate cube roots.
Example
Approximate 330 and also find its value using a calculator.
Answer:
Find the cubes that surround 30.
30 is in between the perfect cubes 27 and 81.
327=3 and 381=4, so 330 is between 3 and 4.
Use a calculator.
330≈3.10723
Try It
[ohm_question]189478[/ohm_question]
Nth Roots
We learned above that the cube root of a number is written with a small number 3, which looks like 3a. This number placed just outside and above the radical symbol and is called the index. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.
We can apply the same idea to any exponent and its corresponding root. The nth root of a is a number that, when raised to the nth power, gives a. For example, 3 is the 5th root of 243 because (3)5=243. If a is a real number with at least one nth root, then the principal nth root of a is the number with the same sign as a that, when raised to the nth power, equals a.
The principal nth root of a is written as na, where n is a positive integer greater than or equal to 2. In the radical expression, n is called the index of the radical.
Definition: Principal nth Root
If a is a real number with at least one nth root, then the principal nth root of a, written as na, is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is n.
Example
Evaluate each of the following:
5−32
481
8−1
Answer:
5−32 Factor 32, which gives (−2)5=−32
481. Factoring can help. We know that 9⋅9=81 and we can further factor each 9: 481=43⋅3⋅3⋅3=434=3
8−1. Since we have an 8th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, −1⋅−1⋅−1⋅−1⋅−1⋅−1⋅−1⋅−1=+1
The steps to consider when simplifying a radical are outlined below.
Simplifying a radical
When working with exponents and radicals:
If n is odd, nxn=x.
If n is even, nxn=∣x∣. (The absolute value accounts for the fact that if x is negative and raised to an even power, that number will be positive, as will the nth principal root of that number.)
In the following video, we show more examples of how to evaluate nth roots.
https://youtu.be/vA2DkcUSRSk
You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can evaluate the radicals 3−81,5−64, and 7−2187 because the all have an odd numbered index, but you cannot evaluate the radicals −100,4−16, or 6−2,500 because they all have an even numbered index.
Try It
[ohm_question]94221[/ohm_question]
Summary
A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property nxn=x if n is odd, and nxn=∣x∣ if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions. Nth roots can be approximated using trial and error or a calculator.
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Simplify Cube Roots (Perfect Cube Radicands).Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
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