https://file.aiquickdraw.com/mathbot/file/1726408124987khmjh96q.jpg

The expression you uploaded translates to the following:

\[
\frac{(0.5u^2v^{-1})^{-3} + 2u^{-2}v^3 \cdot 3u^{-2}v^{-6}}{\left(\frac{7}{u^4 + v^5}\right)^{-1}}
\]

We are also asked to evaluate this expression when \( u = -2 \) and \( v = 3 \).

Let's simplify the expression first and then substitute the values for \( u \) and \( v \).

### Step 1: Simplify the numerator
The numerator consists of two parts:
1. \( (0.5u^2v^{-1})^{-3} \)
   - This can be simplified using the property \( (ab)^n = a^n b^n \).
   \[
   (0.5u^2v^{-1})^{-3} = (0.5)^{-3} \cdot (u^2)^{-3} \cdot (v^{-1})^{-3} = 8u^{-6}v^3
   \]

2. \( 2u^{-2}v^3 \cdot 3u^{-2}v^{-6} \)
   - Multiply the coefficients: \( 2 \cdot 3 = 6 \)
   - Combine powers of \( u \) and \( v \):
     \[
     u^{-2} \cdot u^{-2} = u^{-4}, \quad v^3 \cdot v^{-6} = v^{-3}
     \]
   - Thus, this part simplifies to:
     \[
     6u^{-4}v^{-3}
     \]

So, the numerator becomes:
\[
8u^{-6}v^3 + 6u^{-4}v^{-3}
\]

### Step 2: Simplify the denominator
The denominator is \( \left(\frac{7}{u^4 + v^5}\right)^{-1} \), which simplifies to:
\[
u^4 + v^5
\]

### Step 3: Final simplified expression
Thus, the entire expression becomes:
\[
\frac{8u^{-6}v^3 + 6u^{-4}v^{-3}}{u^4 + v^5}
\]

### Step 4: Substitute \( u = -2 \) and \( v = 3 \)
We now substitute \( u = -2 \) and \( v = 3 \) into the simplified expression.

- \( u^{-6} = (-2)^{-6} = \frac{1}{64} \)
- \( u^{-4} = (-2)^{-4} = \frac{1}{16} \)
- \( u^4 = (-2)^4 = 16 \)
- \( v^3 = 3^3 = 27 \)
- \( v^{-3} = 3^{-3} = \frac{1}{27} \)
- \( v^5 = 3^5 = 243 \)

Thus, the numerator becomes:
\[
8 \cdot \frac{1}{64} \cdot 27 + 6 \cdot \frac{1}{16} \cdot \frac{1}{27}
\]
Simplify each term:
- \( 8 \cdot \frac{1}{64} \cdot 27 = \frac{216}{64} = \frac{27}{8} \)
- \( 6 \cdot \frac{1}{16} \cdot \frac{1}{27} = \frac{6}{432} = \frac{1}{72} \)

Thus, the numerator becomes:
\[
\frac{27}{8} + \frac{1}{72}
\]

We need to find a common denominator:
\[
\frac{27}{8} = \frac{243}{72}, \quad \text{so the numerator is:} \quad \frac{243 + 1}{72} = \frac{244}{72}
\]

Now, the denominator is:
\[
16 + 243 = 259
\]

### Step 5: Final value
Thus, the expression becomes:
\[
\frac{\frac{244}{72}}{259} = \frac{244}{72 \cdot 259} = \frac{244}{18648}
\]

Simplifying the fraction:
\[
\frac{244}{18648} = \frac{61}{4662}
\]

So, the value of the expression is \( \frac{61}{4662} \).

---

Would you like a detailed breakdown of any part of this process? Here are some follow-up questions to expand your understanding:

1. How do you simplify expressions involving negative exponents?
2. What are the steps to simplify a complex fraction?
3. How do you apply the laws of exponents when multiplying terms with different bases?
4. Can you identify when it's necessary to find a common denominator in fractional expressions?
5. How do you handle expressions when given specific values for variables?

**Tip:** When simplifying expressions, always apply exponent rules carefully to avoid mistakes with negative exponents and fractions.

Lihtsusta avaldis (0.5u^2v^{-1})^{-3} + 2u^{-2}v^3 * 3u^{-2}v^{-6} / (7 / (u^4 + v^5))^{-1} ja arvuta selle väärtus, kui u = -2 ja v = 3.

Simplify and evaluate the algebraic expression involving exponents and fractions when u = -2 and v = 3. This solution covers exponent rules, fraction simplifications, and substitution techniques, suitable for students in Grades 10-12.

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