To simplify the expression [tex]\((2 + 3\sqrt{3})(3 + 4\sqrt{3})\)[/tex], we need to use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last), to expand the product of these two binomials. We will follow these steps:
1. First terms: multiply the first terms in each binomial.
[tex]\[
2 \times 3 = 6
\][/tex]
2. Outer terms: multiply the outer terms in the binomials.
[tex]\[
2 \times 4\sqrt{3} = 8\sqrt{3}
\][/tex]
3. Inner terms: multiply the inner terms in the binomials.
[tex]\[
3\sqrt{3} \times 3 = 9\sqrt{3}
\][/tex]
4. Last terms: multiply the last terms in each binomial.
[tex]\[
3\sqrt{3} \times 4\sqrt{3} = 12(\sqrt{3} \times \sqrt{3}) = 12 \times 3 = 36
\][/tex]
Now, let's combine all these results:
[tex]\[
6 + 8\sqrt{3} + 9\sqrt{3} + 36
\][/tex]
Next, we will combine like terms. The terms [tex]\(8\sqrt{3}\)[/tex] and [tex]\(9\sqrt{3}\)[/tex] are like terms because they both involve [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
8\sqrt{3} + 9\sqrt{3} = 17\sqrt{3}
\][/tex]
Finally, add the constant terms:
[tex]\[
6 + 36 = 42
\][/tex]
So, the simplified form of the expression [tex]\((2 + 3\sqrt{3})(3 + 4\sqrt{3})\)[/tex] is:
[tex]\[
42 + 17\sqrt{3}
\][/tex]
Therefore, the radicals and constants can be expressed as follows:
[tex]\[
\boxed{42 + 17\sqrt{3}}
\][/tex]
Thus, our answer is [tex]\(42 + 17\sqrt{3}\)[/tex].
