Simplify square root of 1+(x^3-1/(4x^3))^2

Problem

√(,1+(x3−1/(4*x3))2)

Solution

1. Expand the squared binomial inside the square root using the identity (a−b)2=a2−2*a*b+b2

(x3−1/(4*x3))2=(x3)2−2*(x3)*(1/(4*x3))+(1/(4*x3))2

2. Simplify the terms within the expansion, noting that the x3 terms in the middle product cancel out.

(x3)2−2/4+(1/(4*x3))2=x6−1/2+1/(16*x6)

3. Substitute the expanded expression back into the radicand and combine the constant terms.

1+x6−1/2+1/(16*x6)=x6+1/2+1/(16*x6)

4. Recognize that the resulting expression is a perfect square trinomial of the form a2+2*a*b+b2=(a+b)2

x6+1/2+1/(16*x6)=(x3+1/(4*x3))2

5. Simplify the square root of the squared expression, assuming the expression is positive.

√(,(x3+1/(4*x3))2)=x3+1/(4*x3)

Final Answer

√(,1+(x3−1/(4*x3))2)=x3+1/(4*x3)

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