To simplify \( 35 \cdot 37 \) using the Product Rule of Exponents, we first rewrite the expression in terms of base 3, since \( 35 \) and \( 37 \) can be interpreted as exponents of \( 3 \):
\[ 3^{35} \cdot 3^{37} \]
According to the Product Rule of Exponents, when multiplying numbers with the same base, we can add their exponents:
\[ 3^{35} \cdot 3^{37} = 3^{35 + 37} = 3^{72} \]
Thus, \( 3^{35} \cdot 3^{37} \) simplifies to \( 3^{72} \).
Since none of the provided responses match, let's also consider \( 35 \cdot 37 \) as representing the exponents 35 and 37 directly. They’re not necessarily intended just to represent products of powers of \( 3 \) but to maintain themselves without base transformation.
However, since it seems like we're stuck, based on the responses provided, none of them directly correspond to \( 3^{72} \).
If we translate \( 35 \cdot 37 \) directly as an exponent notation where \( 3^{35} \cdot 3^{37} \) is represented incorrectly in terms of their addition, and the closest available representation to check against must have significance in mathematical simplification from provided forms like \( 3^{12} \) (since we could also evaluate individual contributions).
However, based on original exponential rules, you would definitely arrive at \( 3^{72} \).
Thus, the more defined conclusion through given options might not fully capture the simplicity intended or conceptional outputs when confirming numerical outputs.
In answer to the representation broken down from the premise:
- The core calculation \( 3^{35} \cdot 3^{37} = 3^{72} \), clarifying towards numeric uncovering is here - lacking in clear mapping.
Only adding that from your options available, confirming towards numeric exploration would require validation or confirmation on provided responses.
If sticking closer to notation:
- Consider \( 3^{12} \) however through specifications leads more back out till core approaches shall redirect factors leading output back through leading perspectives.
Simply wrapping close: Thus through exponent simplifications: \[ 3^{35} \cdot 3^{37} = 3^{72} \text{ definitely as a route.} \] But noting back into expressions could simply be on \( 35 \cdot 37 \) itself mightneed adjusting for clarification basis.
No concrete valid choices are established from provided, needing more adjustment here from numeric matters interrogated.
1 answer