how to work these four problems by simplifying radicals

Question

3√27x³y⁶

 

3√24x⁴y¹⁰

 

x⁹y⁷2^3÷x²y²2³

 

3√4x⁵y²×3√10x⁴y³

Answer 1

You want to factor the expression under the radical (called the "radicand") and look for squared terms.  For the first problem, the radicand is 27x³y⁶.  Factoring:

 

27 = 9*3 = 3²*3

x³ = x²*x

y⁶ = (y³)²

 

The squared terms are highlighted in orange.  Hence:

 

3√27x³y⁶ = 3√{3²*3*x²*x*(y³)²}

 

= 3√3²√3√x²√x√(y³)²

 

= 3*3√3 * x√x * y³

 

= 9xy³√3x

 

Try the other three problems on your own in order to learn how to do it for yourself.