Let's examine the rationality or irrationality of the given expressions step-by-step.
### Expression i: [tex]\((5 + \sqrt{5})(5 + \sqrt{5})\)[/tex]
This expression can be simplified as follows:
1. Recognize it as the square of a binomial:
[tex]\[(5 + \sqrt{5})^2\][/tex]
2. Apply the binomial expansion formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:
[tex]\[(5 + \sqrt{5})^2 = 5^2 + 2 \cdot 5 \cdot \sqrt{5} + (\sqrt{5})^2\][/tex]
3. Simplify each term:
[tex]\[5^2 = 25\][/tex]
[tex]\[2 \cdot 5 \cdot \sqrt{5} = 10\sqrt{5}\][/tex]
[tex]\[(\sqrt{5})^2 = 5\][/tex]
4. Combine the simplified terms:
[tex]\[5^2 + 2 \cdot 5 \cdot \sqrt{5} + (\sqrt{5})^2 = 25 + 10\sqrt{5} + 5\][/tex]
[tex]\[= 30 + 10\sqrt{5}\][/tex]
Since [tex]\(10\sqrt{5}\)[/tex] is irrational (because the product of a rational number (10) and an irrational number ([tex]\(\sqrt{5}\)[/tex]) is irrational), the sum [tex]\(30 + 10\sqrt{5}\)[/tex] is also irrational.
So, the number [tex]\((5 + \sqrt{5})(5 + \sqrt{5})\)[/tex] simplifies to approximately [tex]\(\approx 52.3606797749979\)[/tex] and is irrational.
### Expression ii: [tex]\((5 + \sqrt{5})(5 - \sqrt{5})\)[/tex]
This expression can be simplified using the difference of squares formula:
1. Recognize it as a product of conjugates:
[tex]\[(5 + \sqrt{5})(5 - \sqrt{5})\][/tex]
2. Apply the difference of squares formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]:
[tex]\[(5 + \sqrt{5})(5 - \sqrt{5}) = 5^2 - (\sqrt{5})^2\][/tex]
3. Simplify each term:
[tex]\[5^2 = 25\][/tex]
[tex]\[(\sqrt{5})^2 = 5\][/tex]
4. Combine the simplified terms:
[tex]\[5^2 - (\sqrt{5})^2 = 25 - 5 = 20\][/tex]
The number 20 is rational because it can be expressed as a ratio of two integers (in this case, 20/1).
So, the number [tex]\((5 + \sqrt{5})(5 - \sqrt{5})\)[/tex] simplifies to 20 and is rational.
### Summary
- [tex]\((5 + \sqrt{5})(5 + \sqrt{5})\)[/tex] is irrational.
- [tex]\((5 + \sqrt{5})(5 - \sqrt{5})\)[/tex] is rational.
