Rewrite $\sqrt[3]{4} \cdot \sqrt{2}$ as a single radical.
Answer (2)
To rewrite 3 4 ⋅ 2 as a single radical, we express it as 6 128 . This is achieved by combining their fractional exponents and simplifying. The transformation involved rewriting the radicals in exponential form and adding exponents appropriately.
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To rewrite 3 4 ⋅ 2 as a single radical, we need to understand how to combine the two expressions under one radical. The key here is to express both radicals with a common index.
Express the radicals with an exponent form :
3 4 can be expressed as 4 1/3 .
2 can be expressed as 2 1/2 .
Combine the radicals :
We multiply these two expressions together: 4 1/3 ⋅ 2 1/2 .
Express 4 as a power of 2 :
Notice that 4 = 2 2 . Therefore, 4 1/3 = ( 2 2 ) 1/3 = 2 2/3 .
Combine using the same base :
Now you have 2 2/3 ⋅ 2 1/2 . When multiplying with the same base, you add the exponents: 2 ( 2/3 + 1/2 ) .
Add the exponents :
Convert the fractions to have a common denominator: 2/3 = 4/6 and 1/2 = 3/6 .
Then add the exponents: 4/6 + 3/6 = 7/6 .
Final expression :
Therefore, the expression becomes 2 7/6 , which can be written as a single radical: 6 2 7 .
So, 3 4 ⋅ 2 can be written as 6 2 7 .
