The number √8 is irrational because __
1. It is a decimal that repeats
2. It is a decimal that terminates
3. It is the square root of a perfect square
4. It is a decimal that does not repeat or terminate
5. It is the square root of a non-perfect square
5. It is the square root of a non-perfect square
The correct answer is option 5. It is the square root of a non-perfect square.
The number √8 is irrational because it cannot be expressed as a fraction or a decimal that terminates or repeats. The square root of 8 cannot be simplified to a whole number or fraction, which makes it irrational.
The correct answer is 4. It is a decimal that does not repeat or terminate.
To understand why √8 is irrational, we need to understand what it means for a number to be irrational. Irrational numbers are real numbers that cannot be expressed as a fraction or a ratio of two integers. In other words, they cannot be written as terminating or repeating decimals.
To determine if √8 is irrational, we can start by assuming that it is rational. If √8 is rational, it can be expressed as a fraction in the form of a/b, where a and b are integers with no common factors other than 1, and b is not zero.
Let's take the square of both sides of the equation √8 = a/b to get rid of the square root symbol:
(√8)^2 = (a/b)^2
8 = a^2/b^2
8b^2 = a^2
From this equation, we can see that a^2 must be a multiple of 8, since it is equal to 8 multiplied by b^2. This means that a must be a multiple of 2, since it is squared. Therefore, we can write a = 2k, where k is an integer.
Substituting this into the equation, we get:
8b^2 = (2k)^2
8b^2 = 4k^2
2b^2 = k^2
Here, we can see that k^2 must be a multiple of 2, so k must also be a multiple of 2. Let's write k = 2m, where m is an integer.
Substituting this into the equation, we get:
2b^2 = (2m)^2
2b^2 = 4m^2
b^2 = 2m^2
Now, we have shown that both a and b must be even, since they are both equal to multiples of 2. However, this means that a and b have a common factor of 2, which contradicts our assumption that a/b has no common factors other than 1. Therefore, our assumption that √8 is rational is false.
Since √8 cannot be expressed as a fraction, it is irrational. Additionally, we can notice that √8 is a decimal that neither repeats nor terminates, further confirming that it is irrational.