https://file.aiquickdraw.com/mathbot/file/17369526873052g8m2j21.jpg

The problem from the image reads:

\[
\left(2^{\frac{1}{2}}\right)^{-3} - \left(5 \cdot 2^{-1}\right)^{-3} + \left(1 \cdot 3^{-1}\right)^{-2} - 2^{-4} = ?
\]

We will simplify each term step by step:

---

### Step 1: Simplify \( \left(2^{\frac{1}{2}}\right)^{-3} \)
\[
2^{\frac{1}{2}} = \sqrt{2}, \text{ so } \left(2^{\frac{1}{2}}\right)^{-3} = \left(\sqrt{2}\right)^{-3} = \frac{1}{(\sqrt{2})^3}.
\]
Since \((\sqrt{2})^3 = \sqrt{8}\), this simplifies to:
\[
\left(2^{\frac{1}{2}}\right)^{-3} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}.
\]

---

### Step 2: Simplify \( \left(5 \cdot 2^{-1}\right)^{-3} \)
\[
5 \cdot 2^{-1} = \frac{5}{2}, \text{ so } \left(5 \cdot 2^{-1}\right)^{-3} = \left(\frac{5}{2}\right)^{-3} = \left(\frac{2}{5}\right)^3.
\]
\[
\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}.
\]

---

### Step 3: Simplify \( \left(1 \cdot 3^{-1}\right)^{-2} \)
\[
1 \cdot 3^{-1} = \frac{1}{3}, \text{ so } \left(1 \cdot 3^{-1}\right)^{-2} = \left(\frac{1}{3}\right)^{-2} = \left(3\right)^2 = 9.
\]

---

### Step 4: Simplify \( 2^{-4} \)
\[
2^{-4} = \frac{1}{2^4} = \frac{1}{16}.
\]

---

### Combine All Terms
Substitute all simplified terms into the equation:
\[
\frac{\sqrt{2}}{4} - \frac{8}{125} + 9 - \frac{1}{16}.
\]

We now calculate this step by step:

1. Combine constants \( \frac{\sqrt{2}}{4}, \frac{8}{125}, \frac{1}{16}, \text{ and } 9 \):
   - Convert all terms into a common denominator.
2. Proceed computation sympy import Rational, sqrt

# Define terms
term1 = sqrt(2) / 4
term2 = Rational(8, 125)
term3 = 9
term4 = Rational(1, 16)

# Combine all terms
result = term1 - term2 + term3 - term4
result.evalf()  # Decimal approximationThe simplified result of the given expression is approximately \( 9.227 \). If you would like, I can break down any of the steps further or verify specific parts of the calculation. 

### Follow-up Questions:
1. Would you like a more detailed explanation for one of the terms?
2. Shall I provide a fully symbolic answer without approximations?
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4. Would you like to see similar problems solved?
5. Would visualizations of the simplifications help?

**Tip:** When dealing with negative exponents, always rewrite them as reciprocals for better clarity during simplifications!

Simplify the expression: (2^(1/2))^-3 - (5 * 2^-1)^-3 + (1 * 3^-1)^-2 - 2^-4.

Step-by-step solution to simplify the expression (2^(1/2))^-3 - (5 * 2^-1)^-3 + (1 * 3^-1)^-2 - 2^-4. This solution involves the laws of exponents, fractional powers, and basic arithmetic simplifications. Ideal for Grades 9-12.

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