Final answer:
To prove 7 + 2√3 is an irrational number, one can start by assuming it's rational and find a contradiction. Since √3 is irrational and cannot be expressed as a fraction, any expression that combines √3 with rational numbers like 7 will itself be irrational.
Step-by-step explanation:
The question asks for a proof to show that 7 + 2√3 is an irrational number. An irrational number is a number that cannot be expressed as a fraction a/b, where a and b are integers and b is not zero. To prove this, we can use the fact that the square root of a non-perfect square is irrational, and any sum or difference involving an irrational number and a rational number remains irrational.
Lets assume, for contradiction, that 7 + 2√3 is rational. This would mean that there exist integers p and q (with q not equal to zero) such that:
7 + 2√3 = p/q
By rearranging the equation we can isolate the square root term:
2√3 = p/q - 7
√3 = (p/q - 7)/2
At this point, we'd have an expression stating that √3, which is a known irrational number, is equal to a rational number which is a contradiction. This contradiction implies our initial assumption that 7 + 2√3 is rational is false. Therefore, we can conclude that 7 + 2√3 is an irrational number.