Evaluate fourth root of 147-4 square root of 192

Problem

√(4,147−4√(,192))

Solution

1. Simplify the inner square root by factoring out perfect squares.

√(,192)=√(,64⋅3)

√(,192)=8√(,3)

2. Substitute the simplified square root back into the expression.

√(4,147−4*(8√(,3)))

√(4,147−32√(,3))

3. Rewrite the fourth root as a nested square root.

√(,√(,147−32√(,3)))

4. Transform the inner radicand into the form a−2√(,b) to find a perfect square.

√(,147−2*(16√(,3)))

√(,147−2√(,16⋅3))

√(,147−2√(,256⋅3))

√(,147−2√(,768))

5. Identify two numbers x and y such that x+y=147 and x*y=768

x=144

y=3

6. Simplify the inner square root using the identity √(,x+y−2√(,x*y))=√(,x)−√(,y)

√(,147−2√(,768))=√(,144)−√(,3)

12−√(,3)

7. Substitute this result back into the remaining outer square root.

√(,12−√(,3))

Final Answer

√(4,147−4√(,192))=√(,12−√(,3))

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