What is the simplified form of [tex]\sqrt{\frac{72 x^{16}}{50 x^{36}}}[/tex]? Assume [tex]x \neq 0[/tex].

A. [tex]\frac{6}{5 x^{10}}[/tex]
B. [tex]\frac{6}{5 x^2}[/tex]
C. [tex]\frac{6}{5} x^{10}[/tex]
D. [tex]\frac{6}{5} x^2[/tex]



Answer :

To simplify the expression [tex]\(\sqrt{\frac{72 x^{16}}{50 x^{36}}}\)[/tex], let's break it down step-by-step:

1. Simplify the fraction inside the square root:
[tex]\[ \frac{72 x^{16}}{50 x^{36}} \][/tex]
We can rewrite this as:
[tex]\[ \frac{72}{50} \cdot \frac{x^{16}}{x^{36}} \][/tex]

2. Simplify the constants:
[tex]\[ \frac{72}{50} \][/tex]
Simplify [tex]\(\frac{72}{50}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives:
[tex]\[ \frac{72}{50} = \frac{36}{25} \][/tex]

3. Simplify the variable terms:
[tex]\[ \frac{x^{16}}{x^{36}} \][/tex]
Using the properties of exponents, [tex]\(\frac{x^a}{x^b} = x^{a - b}\)[/tex]:
[tex]\[ \frac{x^{16}}{x^{36}} = x^{16 - 36} = x^{-20} \][/tex]

4. Combine the simplified constants and variable terms:
The expression now becomes:
[tex]\[ \sqrt{\frac{36}{25} \cdot x^{-20}} \][/tex]

5. Distribute the square root:
[tex]\[ \sqrt{\frac{36}{25} \cdot x^{-20}} = \sqrt{\frac{36}{25}} \cdot \sqrt{x^{-20}} \][/tex]

6. Simplify each square root individually:
[tex]\[ \sqrt{\frac{36}{25}} = \frac{\sqrt{36}}{\sqrt{25}} = \frac{6}{5} \][/tex]
For the variable term:
[tex]\[ \sqrt{x^{-20}} = (x^{-20})^{1/2} = x^{-10} \][/tex]

7. Combine the results:
The simplified form is:
[tex]\[ \frac{6}{5} \cdot x^{-10} = \frac{6}{5x^{10}} \][/tex]

Therefore, the simplified form of [tex]\(\sqrt{\frac{72 x^{16}}{50 x^{36}}}\)[/tex] is:

[tex]\[ \frac{6}{5x^{10}} \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{\frac{6}{5x^{10}}} \][/tex]