Question

use a pattern to rewrite the expression, and simplify the result if possible. select the correct choice and, if necessary, fill in the answer box to complete your choice. use the pattern to simplify the expression. then, use properties of exponents to verify the answer. \\(3^2 \cdot 3^9\\) \\(\bigcirc\\) a \\(3^2 \cdot 3^9 = 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2\\), which simplifies to \\(\square\\). (type exponential notation with positive exponents.) \\(\bigcirc\\) b \\(3^2 \cdot 3^9 = (3 \cdot 3 \cdot 3 \cdot 3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3)\\), which cannot be simplified further. \\(\bigcirc\\) c \\(3^2 \cdot 3^9 = 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2\\), which cannot be simplified further. \\(\bigcirc\\) d \\(3^2 \cdot 3^9 = \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}\\), which cannot be simplified further. \\(\bigcirc\\) e \\(3^2 \cdot 3^9 = \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}\\), which simplifies to \\(3 \cdot 3 \cdot 3 \cdot 3\\). (type exponential notation with positive exponents.) \\(\bigcirc\\) f \\(3^2 \cdot 3^9 = (3 \cdot 3 \cdot 3 \cdot 3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3)\\), which simplifies to \\(\square\\). (type exponential notation with positive exponents.)

Explanation:

Step1: Recall exponent product rule

When multiplying exponents with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$

Step2: Apply rule to $3^5 \cdot 3^9$

$3^5 \cdot 3^9 = 3^{5+9}$

Step3: Calculate the sum of exponents

$5+9=14$, so $3^{5+9}=3^{14}$

Step4: Verify with expanded form

$3^5 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$, $3^9 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$. Multiplying them gives $3$ multiplied 14 times, which is $3^{14}$.

Answer:

A. $3^5 \cdot 3^9 = 3^5 \cdot 3^5 \cdot 3^5 \cdot 3^5 \cdot 3^5 \cdot 3^5 \cdot 3^5 \cdot 3^5$, which simplifies to $3^{14}$
(Note: The expanded multiplication in option A has a typo in the number of terms, but it is the only option that follows the correct exponent addition property. The correct simplification result is $3^{14}$)