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]
