To simplify the given expression
[tex]\[ \frac{23 s^5 u t^3}{3 s^2 t}, \][/tex]
we will apply the properties of exponents and basic arithmetic.
### Step-by-Step Solution:
1. Separate the fraction:
[tex]\[ \frac{23 s^5 u t^3}{3 s^2 t} = \frac{23}{3} \cdot \frac{s^5}{s^2} \cdot \frac{u}{1} \cdot \frac{t^3}{t}. \][/tex]
2. Simplify each component:
- The constants:
[tex]\[ \frac{23}{3} \][/tex]
is already in its simplest form.
- The variable [tex]\(s\)[/tex]:
[tex]\[ \frac{s^5}{s^2} \text{ can be simplified using the property of exponents }\left( \frac{a^m}{a^n} = a^{m-n} \right): \][/tex]
[tex]\[ \frac{s^5}{s^2} = s^{5-2} = s^3. \][/tex]
- The variable [tex]\(u\)[/tex]:
[tex]\[ \frac{u}{1} = u. \][/tex]
- The variable [tex]\(t\)[/tex]:
[tex]\[ \frac{t^3}{t} \text{ can be simplified using the same property of exponents:} \][/tex]
[tex]\[ \frac{t^3}{t} = t^{3-1} = t^2. \][/tex]
3. Combine all the simplified components:
[tex]\[ \frac{23}{3} \cdot s^3 \cdot u \cdot t^2. \][/tex]
Putting it all together gives us:
[tex]\[ \frac{23 s^3 u t^2}{3}. \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \boxed{\frac{23 s^3 u t^2}{3}}. \][/tex]
