Certainly! Let's write [tex]\( \left(3^6 \times 3^5\right) \div 3^7 \)[/tex] in the form [tex]\( n:1 \)[/tex], where [tex]\( n \)[/tex] is an integer.
To do this, we will use the properties of exponents. Specifically, we will use the property that states [tex]\( a^m \times a^n = a^{m+n} \)[/tex] and the property that [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex].
1. Combine the product of the exponents:
[tex]\[
3^6 \times 3^5
\][/tex]
According to the exponent multiplication property:
[tex]\[
3^6 \times 3^5 = 3^{6+5} = 3^{11}
\][/tex]
2. Now carry out the division with the exponent of the base 3:
[tex]\[
\frac{3^{11}}{3^7}
\][/tex]
According to the exponent division property:
[tex]\[
\frac{3^{11}}{3^7} = 3^{11-7} = 3^4
\][/tex]
3. Express the result in the required form [tex]\( n:1 \)[/tex]:
[tex]\[
3^4 = 81
\][/tex]
Thus, in the required form [tex]\( n:1 \)[/tex]:
[tex]\[
81:1
\][/tex]
So, [tex]\( n = 81 \)[/tex], the integer value we need.
