https://file.aiquickdraw.com/mathbot/file/172705915350779evexoc.jpg

The problem in the image asks to show that the given values in Mindscapes 10–16 are irrational. Let's break down how to prove a number is irrational for the two examples we can see:

1. **Mindscape 10: \( \sqrt{5} \)**  
2. **Mindscape 11: \( 2 \cdot \sqrt{3} \)**

### How to Prove a Number is Irrational

To prove a number is irrational, we need to demonstrate that it cannot be written as a ratio of two integers. This is done through a proof by contradiction, which assumes the number is rational and leads to a logical inconsistency.

#### Example 1: \( \sqrt{5} \)
1. Assume \( \sqrt{5} \) is rational.
2. That means \( \sqrt{5} = \frac{p}{q} \), where \( p \) and \( q \) are integers with no common factors, and \( q \neq 0 \).
3. Squaring both sides: \( 5 = \frac{p^2}{q^2} \), which gives \( p^2 = 5q^2 \).
4. Therefore, \( p^2 \) is divisible by 5, meaning \( p \) must be divisible by 5 (because 5 is prime).
5. Let \( p = 5k \) for some integer \( k \). Substituting into the equation:  
   \[
   (5k)^2 = 5q^2 \implies 25k^2 = 5q^2 \implies q^2 = 5k^2.
   \]
6. This implies \( q^2 \) is divisible by 5, so \( q \) must also be divisible by 5.
7. But if both \( p \) and \( q \) are divisible by 5, this contradicts the assumption that \( p \) and \( q \) have no common factors. Thus, \( \sqrt{5} \) is irrational.

#### Example 2: \( 2 \cdot \sqrt{3} \)
1. Assume \( 2 \cdot \sqrt{3} \) is rational.
2. Let \( 2 \cdot \sqrt{3} = \frac{p}{q} \), where \( p \) and \( q \) are integers with no common factors, and \( q \neq 0 \).
3. Solving for \( \sqrt{3} \), we get:  
   \[
   \sqrt{3} = \frac{p}{2q}.
   \]
4. This implies \( \sqrt{3} \) is rational, which is a contradiction because it is well-known that \( \sqrt{3} \) is irrational.
5. Hence, \( 2 \cdot \sqrt{3} \) must also be irrational.

These steps show that both \( \sqrt{5} \) and \( 2 \cdot \sqrt{3} \) are irrational.

Would you like further details or any clarifications?  
Here are 5 related questions you might consider:

1. How would you prove that \( \sqrt{2} \) is irrational?
2. Can every square root of a non-perfect square be irrational?
3. What is the difference between rational and irrational numbers in terms of their decimal expansions?
4. How does the proof for \( \sqrt{5} \) change if you consider \( \sqrt{6} \)?
5. Why does the irrationality of \( \sqrt{n} \) depend on \( n \) being prime or not?

**Tip**: Proof by contradiction is a powerful method in mathematics, especially when proving something like irrationality. Make sure you're comfortable with setting up logical contradictions.

In Mindscapes 10–16, show that the value given is irrational. Example values include sqrt(5) and 2 * sqrt(3).

Learn how to prove that sqrt(5) and 2 * sqrt(3) are irrational using proof by contradiction. This problem involves square roots and irrational number concepts, suitable for high school students.

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