Simplify

Problem

√(,7*x^(−1/2))/(4√(,2)+√(,7*x))

Solution

1. Rewrite the numerator by applying the property of radicals and negative exponents to express x^(−1/2) as 1/√(,x)

√(,7)/√(,√(,x))=√(,7)/(x^(1/4))

2. Substitute the simplified numerator back into the original expression.

√(,7)/(x^(1/4))/(4√(,2)+√(,7*x))

3. Multiply the numerator and the denominator by x^(1/4) to eliminate the complex fraction.

√(,7)/(x^(1/4)*(4√(,2)+√(,7*x)))

4. Distribute x^(1/4) into the terms in the denominator, noting that √(,7*x)=√(,7)*x^(1/2)

√(,7)/(4√(,2)*x^(1/4)+√(,7)*x^(1/2)*x^(1/4))

5. Combine the exponents in the second term of the denominator using the rule x^a*x^b=x^(a+b) where 1/2+1/4=3/4

√(,7)/(4√(,2)*x^(1/4)+√(,7)*x^(3/4))

Final Answer

√(,7*x^(−1/2))/(4√(,2)+√(,7*x))=√(,7)/(4√(,2)*x^(1/4)+√(,7)*x^(3/4))

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