Evaluate square root of 2(8- square root of 7)

Problem

√(,2*(8−√(,7)))

Solution

1. Distribute the constant 2 inside the parentheses to simplify the radicand.

√(,16−2√(,7))

2. Identify the expression as a nested radical of the form √(,a−2√(,b)) To simplify this, we look for two numbers x and y such that x+y=a and x*y=b

x+y=16

x*y=7

3. Solve for x and y by finding factors of 7 that sum to 16 Since 7 is prime, its only integer factors are 7 and 1 However, 7 + 1 = 8,n*o*t6.T*h*i*s(i)*n*d(i)*c*a*t*e*s(t)*h*a*t*t*h*e*e*x*p*r*e*s(s(i))*o*n*d(o)*e*s(n)*o*t*s(i)*m*p*l*i*ƒ*y*i*n*t*o*a*b*a*s(i)*c*ƒ*o*r*msqrt{x} - \sqrt{y}$ using integers.

4. Conclude that the expression is already in its simplest radical form, as there are no perfect square factors to extract and the nested radical cannot be denested into simpler integer-based radicals.

Final Answer

√(,2*(8−√(,7)))=√(,16−2√(,7))

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