Sure, let's break this down step-by-step to simplify the given expression: [tex]\( 3^{n+2} \times 9^n \div 27^n \)[/tex].
1. Rewrite each term using a common base:
- The base for all the terms is 3.
- Rewrite [tex]\( 9^n \)[/tex] as [tex]\( (3^2)^n = 3^{2n} \)[/tex].
- Rewrite [tex]\( 27^n \)[/tex] as [tex]\( (3^3)^n = 3^{3n} \)[/tex].
So the expression becomes:
[tex]\[
\frac{3^{n+2} \times 3^{2n}}{3^{3n}}
\][/tex]
2. Combine the terms in the numerator:
- Use the property of exponents that [tex]\( a^m \times a^n = a^{m+n} \)[/tex].
[tex]\[
3^{n+2} \times 3^{2n} = 3^{(n+2) + 2n} = 3^{n+2+2n} = 3^{3n+2}
\][/tex]
Now the expression is:
[tex]\[
\frac{3^{3n+2}}{3^{3n}}
\][/tex]
3. Simplify the division using properties of exponents:
- Use the property that [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex].
[tex]\[
\frac{3^{3n+2}}{3^{3n}} = 3^{(3n+2) - 3n} = 3^2
\][/tex]
4. Final simplification:
[tex]\[
3^2 = 9
\][/tex]
So, the given expression simplifies to [tex]\( 9 \)[/tex].
