We are given two expressions.
sqrt(76+n) [0.5em]
sqrt(2n+26)
We will find the smallest value of n that makes each number rational. To find n let's recall the generalizations about being a rational number for a square root of a number.
When a number is a perfect square, its square root is a rational number. The square roots of positive integers that are non-perfect squares are irrational numbers.
Therefore, we need to have perfect squares inside the square roots of the given expressions. To this end, we will try to find n that makes both expressions perfect squares. When we look at the first expression, we can see that the number inside the square root is close to 81, which is a perfect square.
81=9* 9
From here we need to find the value of n that makes the expression inside the square root 81.
We found that for n=5 the second expression is also a perfect square.
sqrt(76+ 5) &⇒ sqrt(81)=9 [0.5em]
sqrt(2( 5)+26) &⇒ sqrt(36)=6
Consequently, the smallest value of n that makes each number rational is 5.
