Let's simplify the expression [tex]\(\sqrt{75z} - \sqrt{12z}\)[/tex].
### Step 1: Break Down the Square Roots
First, we simplify each square root term separately.
#### For [tex]\(\sqrt{75z}\)[/tex]:
1. Decompose 75 into its prime factors: [tex]\(75 = 3 \times 5^2\)[/tex].
2. Substitute this into the square root: [tex]\(\sqrt{75z} = \sqrt{3 \times 5^2 \times z}\)[/tex].
3. Apply the square root to [tex]\(5^2\)[/tex]: [tex]\(\sqrt{75z} = 5\sqrt{3z}\)[/tex].
#### For [tex]\(\sqrt{12z}\)[/tex]:
1. Decompose 12 into its prime factors: [tex]\(12 = 4 \times 3 = 2^2 \times 3\)[/tex].
2. Substitute this into the square root: [tex]\(\sqrt{12z} = \sqrt{2^2 \times 3 \times z}\)[/tex].
3. Apply the square root to [tex]\(2^2\)[/tex]: [tex]\(\sqrt{12z} = 2\sqrt{3z}\)[/tex].
### Step 2: Substitute Simplified Terms Back into the Expression
Now we have:
[tex]\[
\sqrt{75z} - \sqrt{12z} = 5\sqrt{3z} - 2\sqrt{3z}
\][/tex]
### Step 3: Combine Like Terms
Since both terms contain a common factor of [tex]\(\sqrt{3z}\)[/tex], we can combine them:
[tex]\[
5\sqrt{3z} - 2\sqrt{3z} = (5 - 2)\sqrt{3z} = 3\sqrt{3z}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{75z} - \sqrt{12z}\)[/tex] is:
[tex]\[
3\sqrt{3z}
\][/tex]
### Final Answer:
[tex]\[
\boxed{3\sqrt{3z}}
\][/tex]
