Find that square root 12 /square root 13 is irrational number
The correct answer and explanation is:
Yes, the expression √12 / √13 is an irrational number.
Explanation:
To understand why √12 / √13 is irrational, start by simplifying the expression: 1213=1213\frac{\sqrt{12}}{\sqrt{13}} = \sqrt{\frac{12}{13}}
Now consider whether the number inside the square root, 12/13, is a perfect square. A number is a perfect square if it can be expressed as the square of a rational number. In this case, 12 and 13 are both integers, but their ratio is not a perfect square. This is because:
- 12 is not a multiple of 13
- 13 is a prime number, and 12 does not include 13 as a factor
- 12/13 is a rational number, but it is not a perfect square
Since 12/13 is not a perfect square, its square root, √(12/13), cannot be a rational number.
In general, the square root of a non-perfect square rational number is always irrational. That means √(12/13) is irrational. Therefore, the original expression √12 / √13, which is equal to √(12/13), is also irrational.
Another way to see this is by considering the definitions:
- A rational number can be expressed as a fraction of two integers a/b, where b is not zero.
- An irrational number cannot be written in that way.
If √12 / √13 were rational, then squaring both sides would give 12/13 as a perfect square, but that is false. So, √12 / √13 is irrational.
Hence, √12 / √13 is an irrational number because it is the square root of a non-perfect square rational number.