Simplify cube root of 54a^7b^4

Problem

√(3,54*a7*b4)

Solution

1. Factor the constant 54 into its prime factors to identify perfect cubes.

54=2⋅27=2⋅3

2. Rewrite the variables a7 and b4 using the largest exponents that are multiples of the index 3

a7=a6⋅a1

b4=b3⋅b1

3. Group all perfect cubes together inside the radical.

√(3,3⋅a6⋅b3⋅2⋅a⋅b)

4. Extract the perfect cubes from the radical by dividing their exponents by 3

√(3,3)=3

√(3,a6)=a2

√(3,b3)=b

5. Simplify the expression by placing the extracted terms outside the radical and leaving the remaining factors inside.

3*a2*b√(3,2*a*b)

Final Answer

√(3,54*a7*b4)=3*a2*b√(3,2*a*b)

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