Question

√180 can be expressed in the form p√q. where p and q are integers. Find the smallest value of p + q. ​

Answer 1

The smallest value of p+q is 11

It happens when p = 6 and q = 5.

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Step-by-step explanation:

Let's factor 180 in such a way that exactly one factor is a perfect square.

I'll ignore the trivial factor of 1.

Here are the possible factorizations we could go with:

180 = 4*45

180 = 9*20

180 = 36*5

Those factorizations then lead to the following

`√(180) = √(4*45) = √(4)*√(45)= 2√(45)\\\\√(180) = √(9*20) = √(9)*√(20)= 3√(20)\\\\√(180) = √(36*5) = √(36)*√(5)= 6√(5)\\\\`

Then we have

p+q = 2+45 = 47

p+q = 3+20 = 23

p+q = 6+5 = 11

The smallest value of p+q is 11 and it happens when p = 6 and q = 5.

Side note: p+q is smallest when we go with the largest perfect square factor.