Use Rational exponents to write as a single radical expression. Assume that all variables represent positive real numbers

Question

Use rational exponents to write as a single radical expression. Assume that all variables represent positive real numbers 

⁵√5 *³√x

Answer 1

Rewrite the radicals as fractional exponents then find a common denominator

 

(5^1/5)(x^1/3)=(5^3/15)(x^5/15)=(5³x⁵)^1/15=¹⁵√(5³x⁵)

Answer 2

⁵√(5)³√(x)

 

To write this expression as a single radical, we must find a way to express the terms with the same type of root so that we may multiply them together. First, we rewrite the terms with fractional exponents.

 

5^1/5x^1/3

 

Given that the denominator represents the type of root, we can multiply the terms by finding the lowest common denominator of their exponents. For 3 and 5 this is 15, so we rewrite both of our terms with exponents that have a denominator of 15.

 

5^3/15x^5/15

 

Now we can switch back to radical notation

 

¹⁵√(5³)¹⁵√(x⁵)

 

With the same type of root we can multiply these

 

¹⁵ √(5³x⁵)

 

And further simplifying the term 5³ gives the answer

 

¹⁵√(125x⁵)