To simplify the expression [tex]\(\sqrt{24 x^7}\)[/tex], we need to factor out any perfect squares from under the square root. Let's break it down step by step.
1. Factor the number 24:
[tex]\[
24 = 4 \cdot 6
\][/tex]
Here, 4 is a perfect square.
2. Rewrite the square root of 24:
[tex]\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2 \sqrt{6}
\][/tex]
3. Consider the variable part [tex]\(x^7\)[/tex]:
We can rewrite [tex]\(x^7\)[/tex] as [tex]\(x^6 \cdot x\)[/tex]. Notice that [tex]\(x^6\)[/tex] is a perfect square because it’s an even power of [tex]\(x\)[/tex], specifically:
[tex]\[
x^6 = (x^3)^2
\][/tex]
Therefore:
[tex]\[
\sqrt{x^7} = \sqrt{x^6 \cdot x} = \sqrt{x^6} \cdot \sqrt{x} = x^3 \cdot \sqrt{x}
\][/tex]
4. Combine both simplified parts:
Substitute back the simplified forms:
[tex]\[
\sqrt{24 x^7} = \sqrt{24} \cdot \sqrt{x^7} = 2 \sqrt{6} \cdot x^3 \cdot \sqrt{x}
\][/tex]
5. Combine the terms involving the square root:
Now we get:
[tex]\[
2 \sqrt{6} \cdot x^3 \cdot \sqrt{x} = 2 x^3 \sqrt{6} \cdot \sqrt{x}
\][/tex]
6. Finally, combine [tex]\(\sqrt{6}\)[/tex] and [tex]\(\sqrt{x}\)[/tex] under a single square root:
Since both are under the square root, they can be combined:
[tex]\[
2 x^3 \sqrt{6x}
\][/tex]
So, the simplest form of [tex]\(\sqrt{24 x^7}\)[/tex] is:
[tex]\[
2 x^3 \sqrt{6x}
\][/tex]
