Prove that 1+√2+√7+√21 is irrational

We want to prove that the expression \( 1 + \sqrt{2} + \sqrt{7} + \sqrt{21} \) is irrational. To do this, we'll assume the opposite — that it is rational — and derive a contradiction.

### Step 1: Assume Rationality
Assume \( 1 + \sqrt{2} + \sqrt{7} + \sqrt{21} = r \), where \( r \) is a rational number.

\[
1 + \sqrt{2} + \sqrt{7} + \sqrt{21} = r
\]

This assumption implies that \( \sqrt{2} + \sqrt{7} + \sqrt{21} = r - 1 \), which is still rational because \( r - 1 \) is rational (since \( r \) and \( 1 \) are rational).

Thus, we have:

\[
\sqrt{2} + \sqrt{7} + \sqrt{21} = r - 1
\]

### Step 2: Square Both Sides
Now, square both sides of this equation:

\[
(\sqrt{2} + \sqrt{7} + \sqrt{21})^2 = (r - 1)^2
\]

Expanding the left-hand side:

\[
(\sqrt{2} + \sqrt{7} + \sqrt{21})^2 = 2 + 7 + 21 + 2\sqrt{14} + 2\sqrt{42} + 2\sqrt{147}
\]

Simplifying the constants:

\[
2 + 7 + 21 = 30
\]

So, the left-hand side becomes:

\[
30 + 2\sqrt{14} + 2\sqrt{42} + 2\sqrt{147}
\]

Expanding the right-hand side:

\[
(r - 1)^2 = r^2 - 2r + 1
\]

Thus, we have the equation:

\[
30 + 2\sqrt{14} + 2\sqrt{42} + 2\sqrt{147} = r^2 - 2r + 1
\]

### Step 3: Separate Rational and Irrational Parts
On the left-hand side, the terms \( 2\sqrt{14} \), \( 2\sqrt{42} \), and \( 2\sqrt{147} \) are irrational numbers, while the constant \( 30 \) is rational. On the right-hand side, \( r^2 - 2r + 1 \) is a rational number since it is formed from \( r \), which we assumed to be rational.

For the equation to hold, the irrational parts on the left-hand side must cancel out, which would imply that:

\[
2\sqrt{14} + 2\sqrt{42} + 2\sqrt{147} = 0
\]

This can only happen if each of these terms individually equals zero. However:

- \( \sqrt{14} \neq 0 \)
- \( \sqrt{42} \neq 0 \)
- \( \sqrt{147} \neq 0 \)

Thus, the irrational terms cannot cancel out.

### Step 4: Contradiction
Since the irrational parts cannot cancel, our initial assumption that \( 1 + \sqrt{2} + \sqrt{7} + \sqrt{21} \) is rational must be false.

### Conclusion
Therefore, the expression \( 1 + \sqrt{2} + \sqrt{7} + \sqrt{21} \) is irrational.

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:

1. Can you prove that \( \sqrt{2} + \sqrt{3} \) is irrational?
2. How would you show that \( \sqrt{5} + \sqrt{7} \) is irrational?
3. What conditions must hold for the sum of two square roots to be rational?
4. Can the sum of more than two square roots ever be rational?
5. How do you prove that \( 1 + \sqrt{3} + \sqrt{5} \) is irrational?

**Tip**: When proving irrationality, assuming the expression is rational and reaching a contradiction with irrational components is a common approach.

This step-by-step guide proves that the expression 1 + √2 + √7 + √21 is irrational using algebraic methods and a proof by contradiction. Perfect for advanced high school students in Grades 11-12.

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