To evaluate numeric numeric values with rational exponents, we follow the steps given below.
Step 1 :
Express the base in exponential form.
Step 2 :
If we have power raised to another power, we will multiply the powers.
Step 3 :
Do possible simplification.
Evaluate the following :
Example 1 :
81^{1/2}
Solution :
= 81^{1/2}
Base = 81 and exponent = 1/2
81 = 9^{2}
81^{1/2 }= (9^{2})^{1/2}
= 9^{(2 x 1/2)}
81^{1/2 }= 9
Example 2 :
16^{3/2}
Solution :
= 16^{3/2}
Base = 16, exponent = 3/2
16 = 4^{2}
= 4^{(}^{2 x }^{3/2)}
= 4^{3}
16^{3/2 }= 64
Example 3 :
10000^{3/4}
Solution :
= 10000^{3/4}
Base = 10000 and exponent = 3/4
10000 = 10^{4}
= 10^{(}^{4 x }^{3/4)}
= 10^{3}
= 1000
Example 4 :
64^{2/3}
Solution :
= 64^{2/3}
Base = 64 and exponent = 2/3
64 = 4^{3}
= 4^{(3 x 2/3)}
= 4^{2}
= 16
Example 5 :
27^{2/3}
Solution :
= 27^{2/3}
Base = 27 and exponent = 2/3
27 = 3^{3}
= 3^{(3 x 2/3)}
= 3^{2}
= 9
Example 6 :
81^{-3/2}
Solution :
= 81^{-3/2}
Base = 81 and exponent = -3/2
81 = 9^{2}
= 9 ^{2 x (-3/2)}
= 9^{-3}
= 1/9^{3}
= 1/729
Example 7 :
If c^{2/5} = 4, then c = ?
Solution :
c^{2/5} = 4
Raising power 5 on both sides.
(c^{2/5})^{5} = 4^{5}
c^{(2/5) x 5 }= 4^{5}
c^{2 }= 4^{5}
Take square roots on both sides.
c = √4^{5}
c = 4x4√4
c = 16√(2x2)
c = 16(2)
c = 32
Example 8 :
27^{4/x }= 81, x = ?
Solution :
27^{4/x }= 81
Try to express the bases 27 and 81 as a multiple of 3.
3^{3} = 27 and 3^{4} = 81
3^{3(4/x) }= 3^{4}
3^{12/x} = 3^{4}
Since the bases are equal, we can equate the powers.
12/x = 4
Take reciprocal on both sides.
x/12 = 1/4
Multiply 12 on both sides.
x = 12/4
x = 3
Example 9 :
20^{1/2 }⋅ 20^{1/2}
Solution :
20^{1/2 }⋅ 20^{1/2}
Using the property a^{m }⋅ a^{n} = a^{m+n}
= 20^{1/2 }⋅ 20^{1/2}
= 20^{(1/2 + 1/2)}
= 20
Example 10 :
5^{1/3} ⋅ 25^{1/3}
Solution :
= 5^{1/3} ⋅ 25^{1/3}
25 = 5^{2}
= 5^{1/3} ⋅ (5^{2})^{1/3}
= 5^{1/3} ⋅ 5^{2/3}
= 5^{(1+2)/3}
= 5^{3/3}
= 5
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Jan 17, 22 10:45 AM
Trigonometry Word Problems Worksheet with Answers
Jan 17, 22 10:41 AM
Trigonometry Word Problems with Solutions
Jan 16, 22 11:56 PM
Writing Numbers in Words Worksheet