Final answer:
The expression 2 + 4√2 is irrational because it combines a rational number (2) with an irrational number (4√2). The presence of the irrational term, the square root of 2, ensures that the sum cannot be expressed as a simple fraction of two integers.
Step-by-step explanation:
To determine whether the expression 2 + 4√2 is rational, we need to understand the nature of each component within it. First, rational numbers are numbers that can be expressed as the quotient of two integers, where the divisor is not zero. The number 2 is rational because it can be expressed as 2/1.
However, the term 4√2 is irrational. The square root of 2 is an irrational number, which means it cannot be expressed as a fraction of two integers. The square root of a non-perfect square is always irrational, and since the square root of 2 cannot be expressed as a fraction, multiplying it by 4 does not change its irrational nature.
When you combine a rational number, which is 2 in this case, with an irrational number, which is 4√2, the result is also irrational. Therefore, 2 + 4√2 is irrational.