Sure! Let's break down the expression step by step:
Given:
[tex]\[ \sqrt{8y} + \sqrt{18y} \][/tex]
where [tex]\( y \)[/tex] is a positive real number.
First, we need to simplify each square root individually.
### Step 1: Simplify [tex]\(\sqrt{8y}\)[/tex]
We can rewrite 8 as [tex]\( 4 \times 2 \)[/tex]:
[tex]\[ \sqrt{8y} = \sqrt{4 \times 2y} \][/tex]
We know that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], so:
[tex]\[ \sqrt{4 \times 2y} = \sqrt{4} \times \sqrt{2y} \][/tex]
Since [tex]\(\sqrt{4} = 2\)[/tex]:
[tex]\[ \sqrt{4} \times \sqrt{2y} = 2 \sqrt{2y} \][/tex]
### Step 2: Simplify [tex]\(\sqrt{18y}\)[/tex]
Similarly, we can rewrite 18 as [tex]\( 9 \times 2 \)[/tex]:
[tex]\[ \sqrt{18y} = \sqrt{9 \times 2y} \][/tex]
Again, using [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{9 \times 2y} = \sqrt{9} \times \sqrt{2y} \][/tex]
Since [tex]\(\sqrt{9} = 3\)[/tex]:
[tex]\[ \sqrt{9} \times \sqrt{2y} = 3 \sqrt{2y} \][/tex]
### Step 3: Combine the simplified expressions
Now we add the simplified square roots together:
[tex]\[ 2 \sqrt{2y} + 3 \sqrt{2y} \][/tex]
Since both terms have the common factor [tex]\(\sqrt{2y}\)[/tex], we can combine them:
[tex]\[ (2 + 3) \sqrt{2y} \][/tex]
[tex]\[ 5 \sqrt{2y} \][/tex]
Therefore, the simplified expression is:
[tex]\[ 5 \sqrt{2} \sqrt{y} \][/tex]
Thus, we have:
[tex]\[ \sqrt{8y} + \sqrt{18y} = 5 \sqrt{2} \sqrt{y} \][/tex]
This is the simplified form of the given expression.
