The product of square root 2 and square root 50 is rational or irrational

The product of square root 2 and square root 50 is rational or irrational

The Correct Answer and Explanation is:

Correct Answer: The product of √2 and √50 is irrational.

Explanation:

To determine whether the product of √2 and √50 is rational or irrational, we first simplify the expression:2×50\sqrt{2} \times \sqrt{50}2​×50​

Using the property of square roots:a×b=ab\sqrt{a} \times \sqrt{b} = \sqrt{ab}a​×b​=ab​

We get:2×50=100\sqrt{2 \times 50} = \sqrt{100}2×50​=100​100=10\sqrt{100} = 10100​=10

This result, 10, is a rational number because it can be expressed as a fraction (10/1), which meets the definition of a rational number: any number that can be written as a ratio of two integers where the denominator is not zero.

So, while √2 and √50 are both irrational numbers individually, their product simplifies in this specific case to a rational number. This is not typical of irrational numbers — usually, multiplying two irrational numbers results in another irrational number. But there are exceptions, like this one.

Let’s break it down further:

• √2 is an irrational number. It cannot be written as a fraction.
• √50 is also irrational, since 50 is not a perfect square. But we can simplify √50 as:

50=25×2=25×2=52\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}50​=25×2​=25​×2​=52​

So now:2×50=2×52=5×2×2=5×4=5×2=10\sqrt{2} \times \sqrt{50} = \sqrt{2} \times 5\sqrt{2} = 5 \times \sqrt{2} \times \sqrt{2} = 5 \times \sqrt{4} = 5 \times 2 = 102​×50​=2​×52​=5×2​×2​=5×4​=5×2=10

Therefore, the final product is 10, which is rational.

So although the original expression involves irrational numbers, the product is a rational number.