Let's go through Octavia's steps and identify where she made an error and how to correct it.
1. Simplification of the Expressions:
- The original expression given is [tex]\((3x^9)^2 (3x^2)^4\)[/tex].
2. Squaring the First Term:
[tex]\[
(3x^9)^2
\][/tex]
This means you need to square both the coefficient and the exponent of [tex]\(x\)[/tex]:
[tex]\[
3^2 \cdot x^{9 \cdot 2} = 9 \cdot x^{18}
\][/tex]
3. Raising the Second Term to the Fourth Power:
[tex]\[
(3x^2)^4
\][/tex]
This means you need to raise both the coefficient and the exponent of [tex]\(x\)[/tex] to the fourth power:
[tex]\[
3^4 \cdot x^{2 \cdot 4} = 81 \cdot x^8
\][/tex]
4. Multiplication of the Simplified Terms:
After simplifying each term, you need to multiply the two results together:
[tex]\[
9 \cdot x^{18} \cdot 81 \cdot x^8
\][/tex]
5. Combining the Coefficients:
Multiply the numerical coefficients:
[tex]\[
9 \cdot 81 = 729
\][/tex]
6. Combining the Exponents of x:
Since the bases are the same (both are [tex]\(x\)[/tex]), you can add the exponents:
[tex]\[
x^{18} \cdot x^8 = x^{18+8} = x^{26}
\][/tex]
7. Final Correct Expression:
Multiplying the numerical coefficients and combining the exponents, we get:
[tex]\[
729 \cdot x^{26}
\][/tex]
Octavia's Error:
1. Octavia incorrectly combined exponents in her first move:
[tex]\[
3^2 \cdot x^{11} \cdot 3^4 \cdot x^6
\][/tex]
This should be [tex]\(x^{18}\)[/tex] and [tex]\(x^8\)[/tex] respectively, instead of [tex]\(x^{11}\)[/tex] and [tex]\(x^6\)[/tex].
2. She correctly multiplied the coefficients but combined the exponents incorrectly:
[tex]\[
3^6 \cdot x^{17}
\][/tex]
She should have added [tex]\(18\)[/tex] and [tex]\(8\)[/tex] (and not [tex]\(11\)[/tex] and [tex]\(6\)[/tex]).
Corrected Work:
[tex]\[
\begin{array}{|c|}
\left(3 x^9\right)^2 (3 x^2)^4 \\
9 x^{18} \cdot 81 x^8 \\
9 \cdot 81 \cdot x^{18 + 8} \\
729 \cdot x^{26} \\
\hline
729 x^{26} \\
\hline
\end{array}
\][/tex]
Thus, the correct simplification of the expression [tex]\((3x^9)^2 (3x^2)^4\)[/tex] is [tex]\(729 x^{26}\)[/tex].
