Sure, let's break it down step by step.
### Problem 1: Simplify [tex]\(\frac{\sqrt{6x^3}}{\sqrt{3x}} \cdot \sqrt{8x^2}\)[/tex]
1. Step 1: Simplify the expression inside the product:
[tex]\[
\frac{\sqrt{6x^3}}{\sqrt{3x}} \cdot \sqrt{8x^2}
\][/tex]
Simplify each square root individually:
[tex]\[
\sqrt{6x^3} = \sqrt{6} \cdot \sqrt{x^3} = \sqrt{6} \cdot x^{3/2}
\][/tex]
[tex]\[
\sqrt{3x} = \sqrt{3} \cdot \sqrt{x} = \sqrt{3} \cdot x^{1/2}
\][/tex]
[tex]\[
\sqrt{8x^2} = \sqrt{8} \cdot \sqrt{x^2} = \sqrt{8} \cdot x = 2\sqrt{2} \cdot x
\][/tex]
2. Step 2: Substitute these simplified expressions into the original:
[tex]\[
\frac{\sqrt{6} \cdot x^{3/2}}{\sqrt{3} \cdot x^{1/2}} \cdot 2\sqrt{2} \cdot x
\][/tex]
3. Step 3: Simplify the fraction:
[tex]\[
\frac{\sqrt{6}}{\sqrt{3}} \cdot \frac{x^{3/2}}{x^{1/2}} = \sqrt{\frac{6}{3}} \cdot x^{3/2 - 1/2} = \sqrt{2} \cdot x
\][/tex]
4. Step 4: Combine all parts:
[tex]\[
\sqrt{2} \cdot x \cdot 2\sqrt{2} \cdot x = 2 \cdot \sqrt{2} \cdot \sqrt{2} \cdot x^2 = 2 \cdot 2 \cdot x^2 = 4x^2
\][/tex]
Thus, [tex]\(\frac{\sqrt{6x^3}}{\sqrt{3x}} \cdot \sqrt{8x^2} = 4x^2\)[/tex].
### Problem 2: Simplify [tex]\(\sqrt{16x^4}\)[/tex]
1. Step 1: Simplify the square root:
[tex]\[
\sqrt{16x^4} = \sqrt{16} \cdot \sqrt{x^4} = 4 \cdot x^2 = 4x^2
\][/tex]
Thus, [tex]\(\sqrt{16x^4} = 4x^2\)[/tex].
### Problem 3: Simplify [tex]\(4x\sqrt{x^3}\)[/tex]
1. Step 1: Simplify the square root:
[tex]\[
4x\sqrt{x^3} = 4x \cdot \sqrt{x^2 \cdot x} = 4x \cdot \sqrt{x^2} \cdot \sqrt{x} = 4x \cdot x \cdot \sqrt{x}
\][/tex]
2. Step 2: Combine the terms:
[tex]\[
4x^2 \cdot \sqrt{x}
\][/tex]
Thus, [tex]\(4x\sqrt{x^3} = 4x^2\sqrt{x}\)[/tex].
### Problem 4: Simplify [tex]\(16x^4\)[/tex]
There is no further simplification needed as it is already in its simplest form.
Thus, [tex]\(16x^4 = 16x^4\)[/tex].
### Problem 5: Simplify [tex]\(4x^2\)[/tex]
There is no further simplification needed as it is already in its simplest form.
Thus, [tex]\(4x^2 = 4x^2\)[/tex].
In summary, the simplified results are:
- [tex]\(\frac{\sqrt{6x^3}}{\sqrt{3x}} \cdot \sqrt{8x^2} = 4x^2\)[/tex]
- [tex]\(\sqrt{16x^4} = 4x^2\)[/tex]
- [tex]\(4x\sqrt{x^3} = 4x^2\sqrt{x}\)[/tex]
- [tex]\(16x^4 = 16x^4\)[/tex]
- [tex]\(4x^2 = 4x^2\)[/tex]
