Answer:
To prove that 2√3 - √5 is an irrational number, we will use proof by contradiction.
**Assume that 2√3 - √5 is a rational number.**
This means that it can be expressed as a fraction of two integers, p/q, where p and q are coprime (have no common factors other than 1).
We can square both sides of the equation to get rid of the radical:
```
(2√3 - √5)^2 = (p/q)^2
This simplifies to:
12 - 4√15 = p^2/q^2
Multiplying both sides by q^2, we get:
12q^2 - 4q^2√15 = p^2
Moving the p^2 term to the right side of the equation, we get:
4q^2√15 = p^2 - 12q^2
Dividing both sides by 4q^2, we get:
√15 = (p^2 - 12q^2) / 4q^2
This means that √15 is rational, which is a contradiction, since √15 is irrational.
**Therefore, our original assumption must have been wrong, and 2√3 - √5 must be irrational.**
Here is a more concise version of the proof:
**Assume that 2√3 - √5 is rational.**
Then it can be written as a fraction p/q, where p and q are coprime. Squaring both sides, we get:
This means that √15 is rational, which is a contradiction. Therefore, 2√3 - √5 must be irrational.
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Answers & Comments
Answer:
To prove that 2√3 - √5 is an irrational number, we will use proof by contradiction.
**Assume that 2√3 - √5 is a rational number.**
This means that it can be expressed as a fraction of two integers, p/q, where p and q are coprime (have no common factors other than 1).
We can square both sides of the equation to get rid of the radical:
```
(2√3 - √5)^2 = (p/q)^2
```
This simplifies to:
```
12 - 4√15 = p^2/q^2
```
Multiplying both sides by q^2, we get:
```
12q^2 - 4q^2√15 = p^2
```
Moving the p^2 term to the right side of the equation, we get:
```
4q^2√15 = p^2 - 12q^2
```
Dividing both sides by 4q^2, we get:
```
√15 = (p^2 - 12q^2) / 4q^2
```
This means that √15 is rational, which is a contradiction, since √15 is irrational.
**Therefore, our original assumption must have been wrong, and 2√3 - √5 must be irrational.**
Here is a more concise version of the proof:
**Assume that 2√3 - √5 is rational.**
Then it can be written as a fraction p/q, where p and q are coprime. Squaring both sides, we get:
```
12 - 4√15 = p^2/q^2
```
Multiplying both sides by q^2, we get:
```
4q^2√15 = p^2 - 12q^2
```
Dividing both sides by 4q^2, we get:
```
√15 = (p^2 - 12q^2) / 4q^2
```
This means that √15 is rational, which is a contradiction. Therefore, 2√3 - √5 must be irrational.