Question

The Real Number System - 1/5
+0
Determine if the answer to the expression below will be rational or irrational.
3square18 + 3square12 + 2square 27
rational or irrational

Answer 1

To determine if the expression \( 3\sqrt{18} + 3\sqrt{12} + 2\sqrt{27} \) is rational or irrational, we will simplify each square root term first.

1. **Simplifying \( \sqrt{18} \)**:
\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
\]

2. **Simplifying \( \sqrt{12} \)**:
\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
\]

3. **Simplifying \( \sqrt{27} \)**:
\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\]

Now, substituting these back into the expression:

\[
3\sqrt{18} + 3\sqrt{12} + 2\sqrt{27} = 3(3\sqrt{2}) + 3(2\sqrt{3}) + 2(3\sqrt{3})
\]

This simplifies to:

\[
9\sqrt{2} + 6\sqrt{3} + 6\sqrt{3}
\]

Combining like terms:

\[
9\sqrt{2} + 12\sqrt{3}
\]

Now, we analyze whether this sum is rational or irrational. The term \( \sqrt{2} \) is irrational and \( \sqrt{3} \) is also irrational.

Any expression that includes the sum of a rational multiple of an irrational number and another rational multiple of a different irrational number (which cannot be expressed as a rational number) will remain irrational. Specifically, \( 9\sqrt{2} \) and \( 12\sqrt{3} \) cannot combine to form a rational number.

Thus, the final result \( 9\sqrt{2} + 12\sqrt{3} \) is irrational.

In conclusion, the expression \( 3\sqrt{18} + 3\sqrt{12} + 2\sqrt{27} \) is **irrational**.