Let's simplify the expression: [tex]\(\left(6^{2n+3}\right)\left(8^n\right)\left(3^{2n}\right)\)[/tex].
Step 1: Rewrite using base factors
- Recognize that [tex]\(6\)[/tex] can be written as [tex]\(2 \times 3\)[/tex]:
[tex]\[
6^{2n+3} = (2 \cdot 3)^{2n+3}
\][/tex]
- Recognize that [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]:
[tex]\[
8^n = (2^3)^n = 2^{3n}
\][/tex]
- The term [tex]\(3^{2n}\)[/tex] remains unchanged.
Step 2: Apply exponent rules
- For the term [tex]\((2 \cdot 3)^{2n+3}\)[/tex], we can apply the rule [tex]\((ab)^m = a^m \cdot b^m\)[/tex]:
[tex]\[
(2 \cdot 3)^{2n+3} = 2^{2n+3} \cdot 3^{2n+3}
\][/tex]
- Now our expression becomes:
[tex]\[
(2^{2n+3} \cdot 3^{2n+3}) \cdot 2^{3n} \cdot 3^{2n}
\][/tex]
Step 3: Combine terms with the same base
- Combine the terms with base [tex]\(2\)[/tex]:
[tex]\[
2^{2n+3} \cdot 2^{3n}
\][/tex]
Using the rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
2^{(2n+3) + 3n} = 2^{5n+3}
\][/tex]
- Combine the terms with base [tex]\(3\)[/tex]:
[tex]\[
3^{2n+3} \cdot 3^{2n}
\][/tex]
Using the same exponent rule:
[tex]\[
3^{(2n+3) + 2n} = 3^{4n+3}
\][/tex]
Step 4: Write the final simplified expression
Therefore, the simplified form of the expression is:
[tex]\[
(2^{5n+3}) \cdot (3^{4n+3})
\][/tex]
So, the final simplified result is:
[tex]\[
(2^{5n+3}) \cdot (3^{4n+3})
\][/tex]
