Write each expression in radical form :
Problem 1 :
m^{3/5}
Solution :
= m^{3/5}
We can write the fractional power as product of integer and fraction. So, we get
= m^{3 ⋅ (1/5)}
Power 1/5 can be used as 5^{th} root.
= ^{5}√m^{3}
Problem 2 :
(10x)^{-3/4}
Solution :
First let us write -3/4 as a product of integer and fraction.
(10x)^{-3/4 }= (10x)^{-3 }^{⋅ }^{(1/4)}
Changing power 1/4 as 4^{th} root.
= ^{4}√(10x)^{-3}
In order to change the negative exponent as positive exponent, we will flip the base.
= ^{4}√1/(10x)^{3}
= 1/^{4}√(10x)^{3}
Problem 3 :
(7x)^{3/2}
Solution :
= (7x)^{3/2}
Writing the fractional power as a product of integer and fraction, we get
= (7x)^{3}^{⋅ }^{(1/2)}
Power 1/2 can be written as square root.
= √(7x)^{3}
Here 7x is repeated 3 times inside the square root.
Accordingly the rule in square root, if we see two same terms inside the square root, we can take one out of the radical.
= √(7x) (7x) (7x)
= 7x√(7x)
Problem 4 :
(6b)^{-4/3}
Solution :
= (6b)^{-4/3}
Writing the fractional power as a product of integer and fraction, we get
= (6b)^{-4}^{⋅ }^{(1/3)}
1/3 can be written as cube root.
= ^{3}√(6b)^{-4}
In order to change the negative exponent as positive exponent, we will flip the base.
= ^{3}√1/(6b)^{4}
Since it is cube root, we take one term out of the cube root for every three same terms.
= ^{3}√1/(6b) ⋅ (6b) ⋅ (6b) ⋅ (6b)
= 1/(6b ^{3}√6b)
Problem 5 :
8^{2/3}
Solution :
Method 1 :
= 8^{2/3}
= 8^{2 ⋅ (1/3) }
Writing 1/3 as cube root.
= ^{3}√8^{2}
= ^{3}√64
= ^{3}√(4 ⋅ 4 ⋅ 4)
= 4
Method 2 :
When we evaluate with just numerical values in the base, we will try to write the base in exponential form.
= 8^{2/3}
Writing 8 in exponential form, we get 8 = 2^{3}
= (2^{3})^{2/3}
When we have power raised to another power, we will multiply both the powers.
= 2^{3 x (}^{2/3)}
= 2^{2}
= 4
Problem 6 :
100^{-3/2}
Solution :
Method 1 :
= 100^{-3/2}
= 100^{-3(1/2)}
= √(100)^{-3}
= 1/√(100)^{3}
= 1/100√100
= 1/100(10)
= 1/1000
Method 2 :
= 100^{-3/2}
Writing 100 in exponential form, we get
100 = 10^{2}
= (10)^{2(}^{-3/2)}
= 10^{-3}
= 1/10^{3}
= 1/1000
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