Question

Explain why V8 is
a rational number, but √8 is not a rational
number.

Answer 1

The question involves a typo; 'V8' should be '8^1/3'. The cube root of 8 (8^1/3) is a rational number because it equals 2, while the square root of 8 (√8) is not rational because it simplifies to 2√2, including the irrational number √2.

Step-by-step explanation:

The student is asking why V8 is considered a rational number, while √8 (sqrt(8)) is not a rational number. To clarify, it seems there is a typo in the question and 'V8' should be '8^1/3' (the cube root of 8), since that would make it a rational number. A rational number is any number that can be expressed as the fraction of two integers. The cube root of 8, or 8^1/3, is 2 because 2 x 2 x 2 equals 8, making it a rational number because it can be expressed as the fraction 2/1.

On the other hand, √8 (sqrt(8)) is not a rational number because it cannot be expressed as a fraction of two integers. The square root of 8 simplifies to 2√2, which includes an irrational component (√2). Irrational numbers are numbers that cannot be exactly expressed as a fraction. As √2 is an irrational number, any multiple of it, like 2√2, is also irrational.

Answer 2

8 is a rational number because it can be expressed as a quotient of two integers. The square root of 8 is an irrational number because it cannot be written in the form p/q (where p and q are both integers) q ≠ 0