What is the simplified form of [tex]\sqrt{\frac{2160 x^8}{60 x^2}}[/tex]? Assume [tex]x \neq 0[/tex].

A. [tex]36 x^3[/tex]
B. [tex]36 x^2[/tex]
C. [tex]6 x^3[/tex]
D. [tex]6 x^2[/tex]



Answer :

To simplify the expression [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex], we can follow these steps systematically:

1. Simplify the fraction inside the square root:

[tex]\[ \frac{2160 x^8}{60 x^2} \][/tex]

First, compute the numerical part:

[tex]\[ \frac{2160}{60} \][/tex]

Dividing 2160 by 60:

[tex]\[ \frac{2160}{60} = 36 \][/tex]

Now, handle the variable part:

[tex]\[ \frac{x^8}{x^2} \][/tex]

Using the properties of exponents, [tex]\(\frac{x^8}{x^2} = x^{8-2} = x^6\)[/tex].

So, the fraction simplifies to:

[tex]\[ \frac{2160 x^8}{60 x^2} = 36 x^6 \][/tex]

2. Simplify the expression under the square root:

[tex]\[ \sqrt{36 x^6} \][/tex]

3. Separate the square root of the product:

Using the property of square roots, [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:

[tex]\[ \sqrt{36 x^6} = \sqrt{36} \cdot \sqrt{x^6} \][/tex]

4. Compute the individual square roots:

[tex]\[ \sqrt{36} = 6 \][/tex]

For the second term, [tex]\( \sqrt{x^6} \)[/tex]:

Recall that [tex]\( \sqrt{x^6} = x^{6/2} = x^3 \)[/tex].

5. Combine the results:

[tex]\[ \sqrt{36 x^6} = 6 \cdot x^3 = 6x^3 \][/tex]

Thus, the simplified form of [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex] is [tex]\(6x^3\)[/tex].

Therefore, the correct answer is:

[tex]\[ \boxed{6 x^3} \][/tex]