To simplify the expression [tex]\(\sqrt{54 z} - \sqrt{24 z}\)[/tex], we'll go through the following steps:
1. Factorize the expressions inside the square roots:
- For [tex]\(\sqrt{54z}\)[/tex], we notice that [tex]\(54\)[/tex] can be factored into [tex]\(9 \times 6\)[/tex].
- For [tex]\(\sqrt{24z}\)[/tex], we notice that [tex]\(24\)[/tex] can be factored into [tex]\(4 \times 6\)[/tex].
2. Simplify each square root separately:
- [tex]\(\sqrt{54z} = \sqrt{9 \times 6 \times z}\)[/tex].
Since [tex]\(\sqrt{9} = 3\)[/tex], this simplifies to [tex]\(3 \sqrt{6z}\)[/tex].
- [tex]\(\sqrt{24z} = \sqrt{4 \times 6 \times z}\)[/tex].
Since [tex]\(\sqrt{4} = 2\)[/tex], this simplifies to [tex]\(2 \sqrt{6z}\)[/tex].
3. Substitute the simplified forms back into the expression:
- [tex]\(\sqrt{54z} - \sqrt{24z} = 3 \sqrt{6z} - 2 \sqrt{6z}\)[/tex].
4. Combine like terms:
- We have [tex]\(3 \sqrt{6z}\)[/tex] and [tex]\(2 \sqrt{6z}\)[/tex] which are like terms.
- Subtract [tex]\(2 \sqrt{6z}\)[/tex] from [tex]\(3 \sqrt{6z}\)[/tex]:
[tex]\[
3 \sqrt{6z} - 2 \sqrt{6z} = (3 - 2) \sqrt{6z} = 1 \sqrt{6z} = \sqrt{6z}.
\][/tex]
So, the simplified form of the expression [tex]\(\sqrt{54z} - \sqrt{24z}\)[/tex] is [tex]\(\sqrt{6z}\)[/tex].
