Final answer:
The expression √3 (5 + √49) simplifies to 6√3, which is an irrational number because it represents the product of a rational number (6) and an irrational number (√3). The nature of multiplication between rational and irrational numbers confirms that their product is irrational.
Step-by-step explanation:
The question asks about the nature of the expression √3 (5 + √49). First, it's important to simplify the expression inside the parentheses. We know that the square root of 49 is a rational number, which is 7. This is because 7 times 7 equals 49. Hence, the expression inside the parentheses becomes 5 + 7, which simplifies to 12. Now, we have the expression √3 times 12.
Since 12 is a rational number and √3 is an irrational number, their product would typically be irrational. However, since in this case 12 is a multiple of 3, we have to consider whether √3 times 12 can be simplified further. The factor of 3 under the square root can be extracted as a 3 outside of the root, giving us the new expression 3√3. We can now simplify further by canceling out one 3 from 12 with the √3, leaving us with the simplified expression 6√3.
Therefore, 6√3 is the product of a rational number, 6, and an irrational number, √3. By definition, such a product is always irrational because you cannot express √3 as a fraction of two integers, and any non-zero rational number multiplied by an irrational number is still irrational. Thus, the correct answer is B) It is irrational because it is the product of a rational number and an irrational number.