Questions: Simplify: 2 √27 - 3 √108 - 8 √108

Solution Steps

To simplify the expression \(2 \sqrt{27} - 3 \sqrt{108} - 8 \sqrt{108}\), we first simplify each square root term. We express the numbers under the square roots as products of perfect squares and simplify. Then, we combine like terms.

We start with the expression \(2 \sqrt{27} - 3 \sqrt{108} - 8 \sqrt{108}\). We simplify each square root term:

1. For \(2 \sqrt{27}\): \[ 2 \sqrt{27} = 2 \sqrt{9 \cdot 3} = 2 \cdot 3 \sqrt{3} = 6 \sqrt{3} \]

2. For \(3 \sqrt{108}\): \[ 3 \sqrt{108} = 3 \sqrt{36 \cdot 3} = 3 \cdot 6 \sqrt{3} = 18 \sqrt{3} \]

3. For \(8 \sqrt{108}\): \[ 8 \sqrt{108} = 8 \sqrt{36 \cdot 3} = 8 \cdot 6 \sqrt{3} = 48 \sqrt{3} \]

Now we substitute the simplified terms back into the expression: \[ 6 \sqrt{3} - 18 \sqrt{3} - 48 \sqrt{3} \] Combining these gives: \[ (6 - 18 - 48) \sqrt{3} = -60 \sqrt{3} \]

Final Answer