4.1RationalExponents
We are familiar with the integer rules for exponents, but can those rules be extended to any rational
¡
exponent? For example, could we figure out what ‘ means? Let’s do some exploring.
Start with the Roots
Let’s start with something that we know such as (√) = . Now let’s rewrite that trying to write the
square root as an exponent. Since we don’t know what power a square root is, we’ll just use a variable for now.
This gives us the following:
(√) = (  ) = Using the same rules as the exponents we know that 2? 1 which means that ? = . We have just proved
that the square root is really the rational exponent of half. Following the same procedure we can see that √ =
¡
¡
¡
 , √ =  , and more generally that √ = ® .
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
Any Rational Exponent
Now that we know that the denominator of an exponent represents what root we are taking, we can then
see what any rational exponent means. For example,
C
= ( C ) = ” C

Also note that due to the commutative property of multiplication, we could have also written this as follows:
C
= ( )C = ( √ )C
So we can see that the numerator of the exponent tells us the power of the number before (or after) we
¯
take the square root. In other words if we have ® where both O and < are integers, then we take to the Oth
power and then take the <th root. We could also take the <th root first and then raise it to the Oth power.
°
®
z = ( √ )° = √ °
®
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Limitations on Rational Exponents
While this process of equivalence seems fairly straight forward, there are some factors to consider. For
example, any rational exponent with a denominator of two won’t work with negative numbers since there is no
square root of negative numbers. In fact this extends to any even root, so for this year we will assume that the
¯
statement ® = ( √ )° = √ ° is true for any non-negative value.
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Evaluating Rational Exponents
Besides just writing out what a rational exponent means (writing an equivalent expression), we can also

just evaluate rational exponents. For example, we could evaluate 16 as follows:
C
16 = √16C = 2C = 8

Notice again that had we worked it out the other way, we would end up with the same solution.
C

16 = ”16C = √4096 = 8
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
Lesson 4.1
Evaluate the following exponents operations giving your answer as a fraction where necessary.

2. 4‘
•
‘
6. 27
1. 4‘

¡
‘
‘
9. 81
4. 9‘

7. 16
‘
5. 8
•
3. 9‘
8. 16

10. 81
•
11. 81
12. 16
Determine if the following equations are true. Justify your answer.
‘

13. 5 = (√5)
¡¡
16. – = √ {
‘
¡¡
19. ? = √? 
•
14. 16 = √16
•

17. ’ = ”s
•
20. @  = √@C
•
15. 160.sD = √16C

¡‘
18. I • = √I
’
•
21. + ¡¡ = √+ s
¡¡
Determine the appropriate exponent to make the equation true.
22. √2C 2•

23. √ s 
25. ”4 ? 4’

28. √6t 6 ?
?
’

’
24. √7{ 7
26. ” ? 

27. ”I ? I •
29. √@ @ ?
30. √3s 3 ?
?

­
?

¡ª
‘
’
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