Let's simplify the expression step-by-step:
[tex]\[
\sqrt{28z} + \sqrt{63z}
\][/tex]
### Step 1: Factor the radicands
We begin by factoring the numbers inside each square root to their prime factors, along with the variable [tex]\( z \)[/tex].
1. For [tex]\( \sqrt{28z} \)[/tex]:
[tex]\[ 28z = 4 \times 7 \times z \][/tex]
So,
[tex]\[ \sqrt{28z} = \sqrt{4 \times 7 \times z} \][/tex]
Since [tex]\(\sqrt{4} = 2\)[/tex]:
[tex]\[ \sqrt{28z} = 2\sqrt{7z} \][/tex]
2. For [tex]\( \sqrt{63z} \)[/tex]:
[tex]\[ 63z = 9 \times 7 \times z \][/tex]
So,
[tex]\[ \sqrt{63z} = \sqrt{9 \times 7 \times z} \][/tex]
Since [tex]\(\sqrt{9} = 3\)[/tex]:
[tex]\[ \sqrt{63z} = 3\sqrt{7z} \][/tex]
### Step 2: Combine like terms
Now we have:
[tex]\[ \sqrt{28z} + \sqrt{63z} = 2\sqrt{7z} + 3\sqrt{7z} \][/tex]
Since both terms have the common factor [tex]\( \sqrt{7z} \)[/tex], we can combine them:
[tex]\[ 2\sqrt{7z} + 3\sqrt{7z} = (2 + 3) \sqrt{7z} \][/tex]
[tex]\[ 2\sqrt{7z} + 3\sqrt{7z} = 5\sqrt{7z} \][/tex]
### Final result:
[tex]\[
\sqrt{28z} + \sqrt{63z} = 5\sqrt{7}\sqrt{z}
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
5\sqrt{7}\sqrt{z}
\][/tex]
