Let's find the simplified form of the expression [tex]\( y = \sqrt{2 x^3} + 5 \)[/tex].
### Step-by-Step Solution:
#### Step 1: Recognize the given function
We start with the function:
[tex]\[ y = \sqrt{2 x^3} + 5 \][/tex]
#### Step 2: Simplify the square root term
First, we focus on the term under the square root. The term [tex]\( 2 x^3 \)[/tex] can be written without changing it, as it is already simplified. We apply the square root to the product inside:
[tex]\[ \sqrt{2 x^3} \][/tex]
#### Step 3: Apply the properties of square roots
Recall that the square root of a product is the product of the square roots. We can separate the square root of the constant and the square root of the variable expression:
[tex]\[ \sqrt{2 x^3} = \sqrt{2} \cdot \sqrt{x^3} \][/tex]
#### Step 4: Simplify [tex]\(\sqrt{x^3}\)[/tex]
The expression [tex]\( x^3 \)[/tex] within the square root can also be simplified. [tex]\(\sqrt{x^3} = \sqrt{x^2 \cdot x} \)[/tex]. Since [tex]\(\sqrt{x^2} = x\)[/tex] for [tex]\( x \geq 0 \)[/tex], we have:
[tex]\[ \sqrt{x^3} = x \cdot \sqrt{x} = x \sqrt{x} \][/tex]
Now combining these simplifications, we get:
[tex]\[ \sqrt{2 x^3} = \sqrt{2} \cdot (x \sqrt{x}) = x \sqrt{2 x} \][/tex]
Combining this with the constant [tex]\( 5 \)[/tex] that is added in the original equation, we get:
[tex]\[ y = \sqrt{2} \cdot x \sqrt{x} + 5 \][/tex]
This can be rewritten as
[tex]\[ y = \sqrt{2} \cdot \sqrt{x^3} + 5 \][/tex]
Which is:
[tex]\[ y = \sqrt{2 x^3} + 5 \][/tex]
So, the final simplified form of the expression is:
[tex]\[ y = \sqrt{2} \cdot \sqrt{x^3} + 5 \][/tex]
Therefore, we can conclude that the simplified form of the expression is:
[tex]\[ y = \sqrt{2} \sqrt{x^3} + 5 \][/tex]
