To simplify the expression [tex]\( 2 \sqrt{50 x^3} + \sqrt{2 x} \)[/tex], we need to break down each radical expression and then combine the like terms.
### Step 1: Simplify [tex]\( 2 \sqrt{50 x^3} \)[/tex]
- First, factor [tex]\( 50 x^3 \)[/tex] inside the square root.
[tex]\[
50 x^3 = 25 \cdot 2 \cdot x^2 \cdot x
\][/tex]
- This allows us to take the square root of the perfect squares [tex]\( 25 \)[/tex] and [tex]\( x^2 \)[/tex]:
[tex]\[
\sqrt{50 x^3} = \sqrt{25 \cdot 2 \cdot x^2 \cdot x} = \sqrt{25} \cdot \sqrt{x^2} \cdot \sqrt{2x} = 5x \sqrt{2x}
\][/tex]
- Now, multiply the expression by the coefficient 2:
[tex]\[
2 \sqrt{50 x^3} = 2 \cdot 5x \sqrt{2x} = 10x \sqrt{2x}
\][/tex]
### Step 2: Simplify [tex]\( \sqrt{2 x} \)[/tex]
- The term [tex]\( \sqrt{2x} \)[/tex] is already in its simplest form.
### Step 3: Combine the radicals
- Both terms [tex]\( 10x \sqrt{2x} \)[/tex] and [tex]\( \sqrt{2x} \)[/tex] share the common radical [tex]\( \sqrt{2x} \)[/tex].
- We can factor out [tex]\( \sqrt{2x} \)[/tex]:
[tex]\[
10x \sqrt{2x} + \sqrt{2x} = \sqrt{2x}(10x + 1)
\][/tex]
### Final Result:
[tex]\[
2 \sqrt{50 x^3} + \sqrt{2 x} = \sqrt{2x}(10x + 1)
\][/tex]
So, the exact simplified expression is:
[tex]\[
\boxed{\sqrt{2x}(10x + 1)}
\][/tex]
