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

Question

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."