Hello there, I'm a math enthusiast with a passion for numbers and their properties. I've spent quite a bit of time exploring the fascinating world of mathematics, and I'm always excited to share my knowledge with others. Today, I'm here to discuss a very interesting question: Is the square root of 64 a rational or irrational number?

Let's dive right into the heart of the matter. A number is considered rational if it can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \) is not zero. An irrational number, on the other hand, is a number that cannot be expressed as a simple fraction; it's a number that goes on forever without repeating.

Now, let's consider the number 64. The square root of 64 is a number that, when multiplied by itself, gives us 64. We can write this mathematically as \( \sqrt{64} = x \), where \( x \times x = 64 \). Solving for \( x \), we find that \( x = 8 \).

So, is 8 a rational number? Yes, it is. The number 8 can be expressed as a fraction \( \frac{8}{1} \), where both the numerator and the denominator are integers, and the denominator is not zero. Therefore, \( \sqrt{64} \) is a rational number.

Now, let's address the reference to the video where Khan explains that \( A \times (\sqrt{8}) \) is irrational. The confusion here might stem from the fact that \( \sqrt{8} \) is an irrational number. The square root of 8 cannot be simplified to a fraction of two integers, which is why it is irrational. However, when you multiply an irrational number by a rational number, the result is not necessarily irrational. It depends on the specific numbers involved.

In the case of \( \sqrt{64} \), we are not dealing with an irrational number being multiplied by another number. Instead, we are simply taking the square root of a perfect square, which is 64. The result of this operation is a whole number, which is rational by definition.

To summarize, the square root of 64 is a rational number because it is equal to 8, and 8 can be expressed as a fraction of two integers. The confusion might arise from the properties of irrational numbers and how they behave when multiplied by other numbers, but in this specific case, we are dealing with a rational result.

I hope this clears up any confusion and provides a comprehensive understanding of the topic. Mathematics is a beautiful subject with many nuances, and I'm always here to help explore and explain them.

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At 4:16 khan explains how [A x (square root of 8) is irrational. But since 'A' is a variable, couldn't 'A' be written as a principle root, like the square root of 8. Because then you would just multiply the two roots, making the square root of 64, which is a rational number.read more >>