Answer:
A proof that square root of 2 is irrational. ... An equation x² = a, and the principal square root ... The number under the radical sign is called the radicand. ... the given square numbers, each product of square numbers is equal to what square ... are relatively prime -- and it will be impossible to divide n· n into m· m and get 2.
Step-by-step explanation:
Rational numbers are numbers that can be represented as a fraction of two integers. For [tex]\sqrt{76 + n}[/tex] and [tex]\sqrt{2n + 26}[/tex] to be rational, the smallest value of n is 5
Given that:
[tex]\sqrt{76 + n}[/tex] and [tex]\sqrt{2n + 26}[/tex] --- correct expressions
For both numbers to be rational, we should be able to represent the numbers as a fraction of two integers.
There are no straight ways to solve this, except trial by error.
When [tex]n = 1[/tex]
[tex]\sqrt{76 + n} = \sqrt{76 + 1} = \sqrt{77} = 8.77496.......[/tex]
The above is not rational
When [tex]n = 2[/tex]
[tex]\sqrt{76 + n} = \sqrt{76 + 2} = \sqrt{78} = 8.83176......[/tex]
The above is not rational
When [tex]n = 3[/tex]
[tex]\sqrt{76 + n} = \sqrt{76 + 3} = \sqrt{79} = 8.88819......[/tex]
The above is not rational
When [tex]n = 4[/tex]
[tex]\sqrt{76 + n} = \sqrt{76 + 4} = \sqrt{80} = 8.944271......[/tex]
The above is not rational
When [tex]n = 5[/tex]
[tex]\sqrt{76 + n} = \sqrt{76 + 5} = \sqrt{81} = 9[/tex]
The above is rational
So, we try [tex]n = 5[/tex] for the second expression
[tex]\sqrt{2n + 26} = \sqrt{2\times 5 + 26} = \sqrt{10 + 26}=\sqrt{36}= 6[/tex]
The above is also rational
Hence, the smallest value of n that makes each of the expression rational is [tex]n = 5[/tex]
Read more about rational numbers at:
https://brainly.com/question/3386568
