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

Problem

√(,1+(x^3−1/(4*x^3))^2)

Solution

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

(x^3−1/(4*x^3))^2=(x^3)^2−2*(x^3)*(1/(4*x^3))+(1/(4*x^3))^2

2. Simplify the terms within the expansion, noting that the x^3 terms in the middle product cancel out.

(x^3)^2−2/4+(1/(4*x^3))^2=x^6−1/2+1/(16*x^6)

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

1+x^6−1/2+1/(16*x^6)=x^6+1/2+1/(16*x^6)

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

x^6+1/2+1/(16*x^6)=(x^3+1/(4*x^3))^2

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

√(,(x^3+1/(4*x^3))^2)=x^3+1/(4*x^3)

Final Answer

√(,1+(x^3−1/(4*x^3))^2)=x^3+1/(4*x^3)

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