Simplify square root of 147m^3n^3

Problem

√(,147*m^3*n^3)

Solution

1. Factor the constant 147 into its prime factors to identify perfect squares.

147=3⋅49

147=3⋅7^2

2. Rewrite the variables m^3 and n^3 as products of perfect squares and remaining factors.

m^3=m^2⋅m

n^3=n^2⋅n

3. Group all perfect square terms together under the radical.

√(,7^2⋅m^2⋅n^2⋅3⋅m⋅n)

4. Apply the product property of radicals √(,a*b)=√(,a)⋅√(,b) to separate the perfect squares.

√(,7^2⋅m^2⋅n^2)⋅√(,3*m*n)

5. Simplify the square root of the perfect squares.

7*m*n√(,3*m*n)

Final Answer

√(,147*m^3*n^3)=7*m*n√(,3*m*n)

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