To express [tex]\( a \)[/tex] in terms of [tex]\( c \)[/tex], we start with the given definitions:
[tex]\[
a = 3b^3 + 2b^6
\][/tex]
[tex]\[
b = (4c + 3)^{1/3}
\][/tex]
First, we need to substitute [tex]\( b \)[/tex] into the expression for [tex]\( a \)[/tex]. Given that [tex]\( b = (4c + 3)^{1/3} \)[/tex], we can find [tex]\( b^3 \)[/tex] and [tex]\( b^6 \)[/tex].
Calculating [tex]\( b^3 \)[/tex]:
[tex]\[
b^3 = \left( (4c + 3)^{1/3} \right)^3 = 4c + 3
\][/tex]
Next, calculating [tex]\( b^6 \)[/tex]:
[tex]\[
b^6 = \left( (4c + 3)^{1/3} \right)^6 = (4c + 3)^2
\][/tex]
Now substitute these results back into the expression for [tex]\( a \)[/tex]:
[tex]\[
a = 3b^3 + 2b^6
\][/tex]
Replace [tex]\( b^3 \)[/tex] and [tex]\( b^6 \)[/tex]:
[tex]\[
a = 3(4c + 3) + 2(4c + 3)^2
\][/tex]
Simplifying further:
[tex]\[
a = 3(4c + 3) + 2(4c + 3)^2
\][/tex]
Finally, write this expression in its simplest form:
[tex]\[
a = 3(4c + 3) + 2(4c + 3)^2
\][/tex]
Thus, the simplified expression for [tex]\( a \)[/tex] in terms of [tex]\( c \)[/tex] is:
[tex]\[
a = 3(4c + 3) + 2(4c + 3)^2
\][/tex]
