Certainly! Let's simplify the expression [tex]\(\sqrt{24 y} + \sqrt{54 y}\)[/tex].
### Step 1: Express Each Term with Simplified Square Roots
First, we express each term under the square root in its simplest radical form.
- For [tex]\(\sqrt{24 y}\)[/tex]:
[tex]\[
24 y = 4 \cdot 6 y = 2^2 \cdot 6 y
\][/tex]
So,
[tex]\[
\sqrt{24 y} = \sqrt{2^2 \cdot 6 y} = \sqrt{2^2} \cdot \sqrt{6 y} = 2 \sqrt{6 y}
\][/tex]
- For [tex]\(\sqrt{54 y}\)[/tex]:
[tex]\[
54 y = 9 \cdot 6 y = 3^2 \cdot 6 y
\][/tex]
So,
[tex]\[
\sqrt{54 y} = \sqrt{3^2 \cdot 6 y} = \sqrt{3^2} \cdot \sqrt{6 y} = 3 \sqrt{6 y}
\][/tex]
### Step 2: Add the Simplified Terms
Now, we add the two simplified terms:
[tex]\[
2 \sqrt{6 y} + 3 \sqrt{6 y}
\][/tex]
### Step 3: Combine Like Terms
Both terms have the common factor [tex]\(\sqrt{6 y}\)[/tex]. We can factor this out:
[tex]\[
2 \sqrt{6 y} + 3 \sqrt{6 y} = (2 + 3) \sqrt{6 y} = 5 \sqrt{6 y}
\][/tex]
### Final Answer
The simplified form of the expression [tex]\(\sqrt{24 y} + \sqrt{54 y}\)[/tex] is:
[tex]\[
5 \sqrt{6 y}
\][/tex]
Therefore, [tex]\(\sqrt{24 y} + \sqrt{54 y} = 5 \sqrt{6} \sqrt{y}\)[/tex].
