To simplify the given expression [tex]\(\sqrt{4 x^2} + \sqrt{16 x^3}\)[/tex] step-by-step, we'll break it down into more manageable parts.
### Step 1: Simplify each square root separately
1. Simplify [tex]\(\sqrt{4 x^2}\)[/tex]:
- Recall that [tex]\(\sqrt{a^2} = a\)[/tex] if [tex]\(a \geq 0\)[/tex].
- Here, we can write [tex]\(4 x^2\)[/tex] as [tex]\((2 x)^2\)[/tex].
- Applying the square root, we get:
[tex]\[
\sqrt{4 x^2} = \sqrt{(2 x)^2} = 2 x
\][/tex]
2. Simplify [tex]\(\sqrt{16 x^3}\)[/tex]:
- We can factor [tex]\(\sqrt{16 x^3}\)[/tex] into simpler components.
- Notice that [tex]\(16 x^3 = 16 x^2 \cdot x\)[/tex].
- We know that [tex]\(\sqrt{16 x^2} = 4 x\)[/tex], because [tex]\(\sqrt{16 x^2} = \sqrt{(4 x)^2} = 4 x\)[/tex].
[tex]\[
\sqrt{16 x^3} = \sqrt{16 x^2 \cdot x} = \sqrt{16 x^2} \cdot \sqrt{x} = 4 x \cdot \sqrt{x} = 4 \sqrt{x^3}
\][/tex]
### Step 2: Combine the simplified expressions
Having simplified both components separately, we can now combine them:
[tex]\[
\sqrt{4 x^2} + \sqrt{16 x^3} = 2x + 4\sqrt{x^3}
\][/tex]
Expressing [tex]\(\sqrt{x^3}\)[/tex] with rational exponents:
[tex]\[
\sqrt{x^3} = x^{3/2}
\][/tex]
So, we have:
[tex]\[
2 x + 4 x^{3/2}
\][/tex]
Therefore, the simplified form of the given expression [tex]\(\sqrt{4 x^2} + \sqrt{16 x^3}\)[/tex] is:
[tex]\[
2 x + 4 x^{3/2}
\][/tex]
For more clarity, this provides a result consistent with the following form:
[tex]\[
2 \sqrt{x^2} + 4 \sqrt{x^3}
\][/tex]
In conclusion, the simplified expression is:
[tex]\[
2\sqrt{x^2} + 4\sqrt{x^3}
\][/tex]
