To simplify the given expression [tex]\(\sqrt{16 a^{12}} b^{20}\)[/tex], let's break it down into manageable steps.
1. Simplify the square root component:
- The term inside the square root is [tex]\(16 a^{12}\)[/tex].
- We know that [tex]\(\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}\)[/tex].
- Therefore, [tex]\(\sqrt{16 a^{12}} = \sqrt{16} \cdot \sqrt{a^{12}}\)[/tex].
- [tex]\(\sqrt{16}\)[/tex] simplifies to 4 because [tex]\(16 = 4^2\)[/tex].
- [tex]\(\sqrt{a^{12}}\)[/tex] simplifies to [tex]\(a^6\)[/tex] because [tex]\(\sqrt{a^n} = a^{n/2}\)[/tex] and [tex]\(12/2 = 6\)[/tex].
- So, [tex]\(\sqrt{16 a^{12}} = 4a^6\)[/tex].
2. Combine with the remaining term:
- Next, we multiply the simplified square root by [tex]\(b^{20}\)[/tex].
- Therefore, [tex]\(4a^6 \cdot b^{20}\)[/tex].
Putting it all together, the simplified form of [tex]\(\sqrt{16 a^{12}} b^{20}\)[/tex] is:
[tex]\[ 4a^6 b^{20} \][/tex]
Comparison with Options:
The options are:
- [tex]\(2 a^3 b^5\)[/tex]
- [tex]\(2 a^3 b^{16}\)[/tex]
- [tex]\(2 a^8 b^5\)[/tex]
- [tex]\(2 a^8 b^{16}\)[/tex]
The simplified expression [tex]\(4a^6 b^{20}\)[/tex] does not match any of the given options exactly. It appears that none of the options provided are correct.
