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Welcome to our Math lesson on Roots as Fractional Powers, this is the second lesson of our suite of math lessons covering the topic of Roots, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Roots as Fractional Powers
From the definition of roots, we have
√x = y ⟹x = yn
This means we can write x as
x = y ∙ y ∙ y…︸n times
Hence, we can express y as
y = x1/n
because multiplying n times both terms above, yields
y ∙ y ∙ y…︸n times = x1/n ∙ x1/n ∙ x1/n…︸n times = x1/n + 1/n +1/n…︸n times = xn/n = x1 = x
Therefore, the two expressions
√x and x1/n
are equivalent (they show the same thing).
In this way, it is easy to conclude that roots are nothing more but simply fractional powers.
Example 3
Calculate the following values:
- 251/2
- 641/3
- (-32)1/5
Solution 3
From the roots - fractional powers equivalence, we have
- 251/2 = √25 = ± 5 because (-5)2 = 25 and ( + 5)2 = 25
- 641/3 = ∛64 = 4 because 43 = 4 ∙ 4 ∙ 4 = 64
- -321/5 = √-32 = -2 because (-2)5 = (-2) ∙ (-2) ∙ (-2) ∙ (-2) ∙ (-2) = -32
More Roots Lessons and Learning Resources
Powers and Roots Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 7.2 | Roots | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 7.2.1 | The Meaning of Roots. Square and Cube Root | | |
| 7.2.2 | Roots as Fractional Powers | | |
| 7.2.3 | Properties of Roots | | |
| 7.2.4 | How to Manually Calculate the Square Root of a Number? | | |
Whats next?
Enjoy the "Roots as Fractional Powers" math lesson? People who liked the "Roots lesson found the following resources useful:
- Fractional Power Feedback. Helps other - Leave a rating for this fractional power (see below)
- Powers and Roots Math tutorial: Roots. Read the Roots math tutorial and build your math knowledge of Powers and Roots
- Powers and Roots Video tutorial: Roots. Watch or listen to the Roots video tutorial, a useful way to help you revise when travelling to and from school/college
- Powers and Roots Revision Notes: Roots. Print the notes so you can revise the key points covered in the math tutorial for Roots
- Powers and Roots Practice Questions: Roots. Test and improve your knowledge of Roots with example questins and answers
- Check your calculations for Powers and Roots questions with our excellent Powers and Roots calculators which contain full equations and calculations clearly displayed line by line. See the Powers and Roots Calculators by iCalculator™ below.
- Continuing learning powers and roots - read our next math tutorial: Standard Form
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