Question

simplify the expression: $3sqrt{54} + 3sqrt{24} - sqrt{6}$

Explanation:

Step1: Simplify each square root term

First, simplify \( \sqrt{54} \). We know that \( 54 = 9\times6 \), so \( \sqrt{54}=\sqrt{9\times6}=\sqrt{9}\times\sqrt{6}=3\sqrt{6} \). Then \( 3\sqrt{54}=3\times3\sqrt{6}=9\sqrt{6} \).

Next, simplify \( \sqrt{24} \). Since \( 24 = 4\times6 \), then \( \sqrt{24}=\sqrt{4\times6}=\sqrt{4}\times\sqrt{6}=2\sqrt{6} \). So \( 3\sqrt{24}=3\times2\sqrt{6}=6\sqrt{6} \).

The term \( \sqrt{6} \) remains as it is.

Step2: Substitute the simplified terms back into the original expression

Substituting, we get:
\( 9\sqrt{6}+6\sqrt{6}-\sqrt{6} \)

Step3: Combine like terms

Now, combine the coefficients of \( \sqrt{6} \). The coefficients are 9, 6, and -1 (since \( -\sqrt{6}=-1\times\sqrt{6} \)).
Adding the coefficients: \( 9 + 6 - 1 = 14 \)
So the expression simplifies to \( 14\sqrt{6} \).

Answer:

\( 14\sqrt{6} \)