Question: User: S^{m+2} _ 3.5^{m+1}
m+2 . S
Top expression: S with superscript m+2, then a long dash-like horizontal line, then 3.5 with superscript m+1; the expression is positioned in the center-top.
Bottom expression: S with superscript m+2, followed by a centered dot/division-like mark, then S; the expression is positioned in the bottom-center.
1. **Stating the problem:** Simplify the expression $$\frac{S^{m+2}}{3.5^{m+1}} \div (S^{m+2} \cdot S)$$.
2. **Rewrite the expression:** The division by a product can be rewritten as multiplication by the reciprocal:
$$\frac{S^{m+2}}{3.5^{m+1}} \times \frac{1}{S^{m+2} \cdot S}$$
3. **Combine the terms:** Multiply the numerators and denominators:
$$\frac{S^{m+2}}{3.5^{m+1}} \times \frac{1}{S^{m+2} \cdot S} = \frac{S^{m+2}}{3.5^{m+1} \cdot S^{m+2} \cdot S}$$
4. **Simplify the powers of $S$:** Using the rule $$a^x \cdot a^y = a^{x+y}$$, the denominator's $S$ terms combine:
$$S^{m+2} \cdot S = S^{m+2+1} = S^{m+3}$$
So the expression becomes:
$$\frac{S^{m+2}}{3.5^{m+1} \cdot S^{m+3}}$$
5. **Cancel common factors:** Using the rule $$\frac{a^x}{a^y} = a^{x-y}$$, we simplify the $S$ terms:
$$\frac{S^{m+2}}{S^{m+3}} = S^{(m+2)-(m+3)} = S^{-1}$$
Intermediate step showing cancellation:
$$\frac{\cancel{S^{m+2}}}{3.5^{m+1} \cdot \cancel{S^{m+2}} \cdot S} = \frac{1}{3.5^{m+1} \cdot S}$$
6. **Final simplified expression:**
$$\frac{1}{3.5^{m+1} \cdot S}$$
**Answer:** $$\boxed{\frac{1}{3.5^{m+1} S}}$$