Let’s solve each problem one by one. Remember: a fractional exponent like $ a^{\frac{m}{n}} $ means you take the
n-th root of $ a $, then raise it to the
m-th power. Or you can do it the other way — raise to the m-th power first, then take the n-th root. Either way works!
We’ll go step by step for each.
---
1. $ 36^{\frac{3}{2}} $
First, take square root (since denominator is 2):
$ \sqrt{36} = 6 $
Then cube it (numerator is 3):
$ 6^3 = 216 $
✔ Answer:
216
---
2. $ 25^{\frac{3}{2}} $
Square root of 25 = 5
Cube it: $ 5^3 = 125 $
✔ Answer:
125
---
3. $ 32^{\frac{3}{5}} $
Fifth root of 32? What number to the 5th power is 32?
$ 2^5 = 32 $ → so fifth root is 2
Now cube it: $ 2^3 = 8 $
✔ Answer:
8
---
4. $ 16^{\frac{5}{4}} $
Fourth root of 16? $ 2^4 = 16 $ → so fourth root is 2
Raise to 5th power: $ 2^5 = 32 $
✔ Answer:
32
---
5. $ 125^{\frac{2}{3}} $
Cube root of 125? $ 5^3 = 125 $ → cube root is 5
Square it: $ 5^2 = 25 $
✔ Answer:
25
---
6. $ 81^{\frac{3}{4}} $
Fourth root of 81? $ 3^4 = 81 $ → fourth root is 3
Cube it: $ 3^3 = 27 $
✔ Answer:
27
---
7. $ 4^{\frac{5}{2}} $
Square root of 4 = 2
Raise to 5th power: $ 2^5 = 32 $
✔ Answer:
32
---
8. $ 8^{\frac{4}{3}} $
Cube root of 8 = 2 (because $ 2^3 = 8 $)
Raise to 4th power: $ 2^4 = 16 $
✔ Answer:
16
---
9. $ 16^{\frac{3}{2}} $
Square root of 16 = 4
Cube it: $ 4^3 = 64 $
✔ Answer:
64
---
10. $ 49^{\frac{1}{2}} $
Square root of 49 = 7
✔ Answer:
7
---
11. $ 64^{\frac{2}{3}} $
Cube root of 64 = 4 (because $ 4^3 = 64 $)
Square it: $ 4^2 = 16 $
✔ Answer:
16
---
12. $ 27^{\frac{2}{3}} $
Cube root of 27 = 3 (because $ 3^3 = 27 $)
Square it: $ 3^2 = 9 $
✔ Answer:
9
---
Final Answer:
36^(3/2) = 216
25^(3/2) = 125
32^(3/5) = 8
16^(5/4) = 32
125^(2/3) = 25
81^(3/4) = 27
4^(5/2) = 32
8^(4/3) = 16
16^(3/2) = 64
49^(1/2) = 7
64^(2/3) = 16
27^(2/3) = 9
Parent Tip: Review the logic above to help your child master the concept of rational exponents worksheet pdf.
