To simplify the expression [tex]\(9 \sqrt{6} - 3 \sqrt{150}\)[/tex], follow these steps:
1. Simplify [tex]\(\sqrt{150}\)[/tex]:
- First, factorize [tex]\(150\)[/tex] into its prime factors: [tex]\(150 = 25 \times 6\)[/tex].
- Recognize that [tex]\(25\)[/tex] is a perfect square: [tex]\(\sqrt{25 \times 6} = \sqrt{25} \cdot \sqrt{6}\)[/tex].
- Calculate [tex]\(\sqrt{25}\)[/tex]: [tex]\(\sqrt{25} = 5\)[/tex], thus [tex]\(\sqrt{150} = 5 \sqrt{6}\)[/tex].
2. Substitute [tex]\(\sqrt{150}\)[/tex] in the original expression:
- Replace [tex]\(\sqrt{150}\)[/tex] with [tex]\(5 \sqrt{6}\)[/tex]:
[tex]\[
9 \sqrt{6} - 3 \cdot 5 \sqrt{6}
\][/tex]
3. Simplify the multiplication:
- Compute [tex]\(3 \cdot 5 \sqrt{6}\)[/tex]:
[tex]\[
3 \cdot 5 \sqrt{6} = 15 \sqrt{6}
\][/tex]
- The expression is now:
[tex]\[
9 \sqrt{6} - 15 \sqrt{6}
\][/tex]
4. Combine like terms:
- Since both terms are multiples of [tex]\(\sqrt{6}\)[/tex], subtract the coefficients:
[tex]\[
(9 - 15) \sqrt{6} = -6 \sqrt{6}
\][/tex]
So, in simplest form, the expression [tex]\(9 \sqrt{6} - 3 \sqrt{150}\)[/tex] simplifies to [tex]\( -6 \sqrt{6} \)[/tex].
