Answer :

Sure, let's simplify the radical [tex]\(\sqrt{48 x^3}\)[/tex] step-by-step.

1. Factor the expression inside the radical:

Start by factoring [tex]\(48 x^3\)[/tex] inside the square root:
[tex]\[ 48 x^3 = (16 \cdot 3) \cdot x^3 = 16 \cdot 3 \cdot x^3 \][/tex]

2. Use the property of square roots:

We can separate the square root of a product into the product of the square roots:
[tex]\[ \sqrt{48 x^3} = \sqrt{16 \cdot 3 \cdot x^3} = \sqrt{16} \cdot \sqrt{3} \cdot \sqrt{x^3} \][/tex]

3. Simplify the square roots of the perfect squares:

[tex]\(\sqrt{16}\)[/tex] is a perfect square, so we simplify it directly:
[tex]\[ \sqrt{16} = 4 \][/tex]

4. Further simplify the remaining square roots:

Now we deal with simplifying [tex]\(\sqrt{x^3}\)[/tex]. We can rewrite [tex]\(x^3\)[/tex] as [tex]\(x^2 \cdot x\)[/tex]:
[tex]\[ \sqrt{x^3} = \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x} = x \cdot \sqrt{x} \][/tex]

5. Combine all simplified parts together:

Now, placing everything back together:
[tex]\[ \sqrt{48 x^3} = 4 \cdot \sqrt{3} \cdot (x \cdot \sqrt{x}) \][/tex]

Finally, combine like terms:
[tex]\[ \sqrt{48 x^3} = 4x \cdot \sqrt{3x} \][/tex]

Thus, the simplified form of [tex]\(\sqrt{48 x^3}\)[/tex] is:
[tex]\[ 4 \sqrt{3} \sqrt{x^3} \][/tex]