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Perform the operations and simplify: [tex]\frac{\sqrt{6 x^3}}{\sqrt{3 x}} \cdot \sqrt{8 x^2}[/tex]

A. [tex]\sqrt{16 x^4}[/tex]
B. [tex]4 x \sqrt{x^3}[/tex]
C. [tex]16 x^4[/tex]
D. [tex]4 x^2[/tex]



Answer :

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]