To simplify the expression [tex]\(\sqrt{144 x^{36}}\)[/tex], let's proceed step by step:
1. Recognize the Square Root Properties:
The square root of a product is the product of the square roots of the individual terms. Thus, we can break it down as follows:
[tex]\[
\sqrt{144 x^{36}} = \sqrt{144} \times \sqrt{x^{36}}
\][/tex]
2. Simplify [tex]\(\sqrt{144}\)[/tex]:
We know that the square root of 144 is 12 because:
[tex]\[
\sqrt{144} = 12
\][/tex]
3. Simplify [tex]\(\sqrt{x^{36}}\)[/tex]:
When dealing with the square root of a power, we can use the property [tex]\(\sqrt{x^n} = x^{n/2}\)[/tex]. Therefore:
[tex]\[
\sqrt{x^{36}} = x^{36/2} = x^{18}
\][/tex]
4. Combine the Results:
Now we can multiply the results from steps 2 and 3:
[tex]\[
\sqrt{144} \times \sqrt{x^{36}} = 12 \times x^{18}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{144 x^{36}}\)[/tex] is:
[tex]\[
12 x^{18}
\][/tex]
Therefore, the correct answer is: [tex]\(12 x^{18}\)[/tex].
