To find the expansion of [tex]\((3c + d^2)^6\)[/tex], we can use the result from expanding this binomial expression. Let's look at each term from the expansion and compare it to the given options:
The expansion is:
[tex]\[729 c^6 + 1458 c^5 d^2 + 1215 c^4 d^4 + 540 c^3 d^6 + 135 c^2 d^8 + 18 c d^{10} + d^{12}.\][/tex]
Let's compare the given options to this expanded form:
1. [tex]\(729 c^6 + 1458 c^5 d^2 + 1215 c^4 d^4 + 540 c^3 d^6 + 135 c^2 d^8 + 18 c d^{10} + d^{12}\)[/tex]
This option matches our expanded form exactly.
2. [tex]\(729 c^6 + 1458 c^5 d + 1215 c^4 d^2 + 540 c^3 d^3 + 135 c^2 d^4 + 18 c d^5 + d^6\)[/tex]
Here, the powers of [tex]\(d\)[/tex] do not match the expanded expression. Specifically, the terms involving [tex]\(d\)[/tex] have incorrect exponents.
3. [tex]\(729 c^6 + 1215 c^5 d^2 + 810 c^4 d^4 + 270 c^3 d^6 + 90 c^2 d^8 + 15 c d^{10} + d^{12}\)[/tex]
This option does not match because the coefficients [tex]\(1458, 540, 135, 18\)[/tex] are incorrect, particularly for the terms [tex]\(c^5 d^2, c^3 d^6, c^2 d^8, c d^{10}\)[/tex].
4. [tex]\(729 c^6 + 243 c^5 d^2 + 81 c^4 d^4 + 27 c^3 d^6 + 9 c^2 d^8 + 3 c d^{10} + d^{12}\)[/tex]
This option has completely incorrect coefficients for all terms except the first and last powers.
5. [tex]\(c^6 + 6 c^5 d^2 + 15 c^4 d^4 + 20 c^3 d^6 + 15 c^2 d^8 + 6 c d^{10} + d^{12}\)[/tex]
This option does not match at all since the coefficients are not consistent with expanding [tex]\((3c + d^2)^6\)[/tex].
Thus, the correct expansion for [tex]\((3c + d^2)^6\)[/tex] is the first option:
[tex]\[729 c^6 + 1458 c^5 d^2 + 1215 c^4 d^4 + 540 c^3 d^6 + 135 c^2 d^8 + 18 c d^{10} + d^{12}.\][/tex]
