https://file.aiquickdraw.com/mathbot/file/17298252453970zqnaw4s.jpg

The mathematical expression in the image is:

\[
\frac{(125)^{\frac{2}{3}} - (25)^{\frac{1}{2}}}{(81)^{\frac{1}{4}} + (27)^{-\frac{1}{3}}} \times \frac{1}{3^{-1}}
\]

Let's solve it step by step.

### Step 1: Simplify the exponents
- \( (125)^{\frac{2}{3}} \): Since \( 125 = 5^3 \), we have:
  \[
  (5^3)^{\frac{2}{3}} = 5^2 = 25
  \]

- \( (25)^{\frac{1}{2}} \): Since \( 25 = 5^2 \), we have:
  \[
  (5^2)^{\frac{1}{2}} = 5
  \]

- \( (81)^{\frac{1}{4}} \): Since \( 81 = 3^4 \), we have:
  \[
  (3^4)^{\frac{1}{4}} = 3
  \]

- \( (27)^{-\frac{1}{3}} \): Since \( 27 = 3^3 \), we have:
  \[
  (3^3)^{-\frac{1}{3}} = 3^{-1} = \frac{1}{3}
  \]

- \( \frac{1}{3^{-1}} \): This simplifies to:
  \[
  3
  \]

### Step 2: Plug in the simplified values
The expression now becomes:

\[
\frac{25 - 5}{3 + \frac{1}{3}} \times 3
\]

### Step 3: Simplify the numerator and denominator
- Numerator: \( 25 - 5 = 20 \)
- Denominator: \( 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \)

So, the expression becomes:

\[
\frac{20}{\frac{10}{3}} \times 3
\]

### Step 4: Simplify the fraction
\[
\frac{20}{\frac{10}{3}} = 20 \times \frac{3}{10} = 6
\]

Finally, multiplying by 3:

\[
6 \times 3 = 18
\]

### Final Answer:
The value of the given expression is **18**.

Would you like further clarification on any step? Here are 5 related questions to explore further:

1. What are the rules for simplifying fractional exponents?
2. How do you simplify negative exponents?
3. Why does \( (a^m)^n = a^{mn} \) when working with powers of powers?
4. How can you apply exponent rules to simplify larger expressions?
5. What are the steps for combining terms with different bases in exponents?

**Tip**: When dealing with complex fractions, simplify each part separately before combining them into a single expression.

What is the value of the following expression: \(\frac{(125)^{\frac{2}{3}} - (25)^{\frac{1}{2}}}{(81)^{\frac{1}{4}} + (27)^{-\frac{1}{3}}} \times \frac{1}{3^{-1}}\)?

Learn how to simplify and solve the expression involving fractional exponents, including step-by-step calculations of powers like (125)^(2/3) and (27)^(-1/3). Ideal for students in Grades 10-12.

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