Question

Solve seven square root three plus two square root nine and explain whether the answer is rational or irrational.

a The answer nine square root of twelve is rational because the sum of a rational number and irrational number is a rational number.
b The answer nine square root of twelve is irrational because the sum of a rational number and irrational number is an irrational number.
c The answer seven square root of three plus six is rational because the sum of a rational number and irrational number is a rational number.
d The answer seven square root of three plus six is irrational because the sum of a rational number and irrational number is an irrational number.

Answer 1

Let's solve the expression \(7\sqrt{3} + 2\sqrt{9}\).

1. **Calculate \( \sqrt{9} \)**:
\[
\sqrt{9} = 3
\]

2. **Substituting \(\sqrt{9}\) into the expression**:
\[
2\sqrt{9} = 2 \times 3 = 6
\]

3. **Now substituting this back into the original expression**:
\[
7\sqrt{3} + 2\sqrt{9} = 7\sqrt{3} + 6
\]

Now we have the expression \(7\sqrt{3} + 6\).

4. **Determine if \(7\sqrt{3} + 6\) is rational or irrational**:
- \(7\sqrt{3}\) is an irrational number because \(\sqrt{3}\) is irrational, and any non-zero rational multiple of an irrational number is also irrational.
- The number \(6\) is rational.
- According to the properties of numbers, the sum of a rational number (6) and an irrational number (\(7\sqrt{3}\)) is an irrational number.

Thus, the final expression \(7\sqrt{3} + 6\) is irrational.

Based on the options provided:
- \(d\) is the correct statement: **"The answer seven square root of three plus six is irrational because the sum of a rational number and an irrational number is an irrational number."**