Simplify fourth root of 16a^4b^5

Problem

√(4,16*a4*b5)

Solution

1. Rewrite the expression using the property of radicals √(n,x*y*z)=√(n,x)⋅√(n,y)⋅√(n,z) to separate the constant and the variables.

√(4,16)⋅√(4,a4)⋅√(4,b5)

2. Evaluate the fourth root of the constant 16 by identifying that 2=16

√(4,16)=2

3. Simplify the fourth root of a4 using the rule √(n,xn)=|x| for even roots, though typically in these contexts variables are assumed positive.

√(4,a4)=a

4. Decompose the term b5 into b4⋅b1 to extract the perfect fourth power.

√(4,b5)=√(4,b4⋅b)

5. Apply the radical property to b4⋅b and simplify.

√(4,b4)⋅√(4,b)=b√(4,b)

6. Combine all the simplified parts back into a single expression.

2⋅a⋅b√(4,b)

Final Answer

√(4,16*a4*b5)=2*a*b√(4,b)

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