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The given expression to simplify is:

\[
\frac{(25)^{\frac{1}{2}} + (36)^{\frac{1}{2}}}{(125)^{\frac{2}{3}} - (27)^{\frac{1}{3}}}
\]

Let's simplify this step by step:

### Step 1: Simplify the square roots and cube roots
- \( (25)^{\frac{1}{2}} = \sqrt{25} = 5 \)
- \( (36)^{\frac{1}{2}} = \sqrt{36} = 6 \)
- \( (125)^{\frac{2}{3}} = \left( 125^{\frac{1}{3}} \right)^2 = (5)^2 = 25 \) (since \(125 = 5^3\))
- \( (27)^{\frac{1}{3}} = \sqrt[3]{27} = 3 \)

### Step 2: Substitute these simplified values into the expression
\[
\frac{5 + 6}{25 - 3} = \frac{11}{22}
\]

### Step 3: Simplify the fraction
\[
\frac{11}{22} = \frac{1}{2}
\]

So, the simplified expression is \( \frac{1}{2} \).

Would you like further clarification or details on any steps? 

Here are 5 related questions:
1. What is the general formula for simplifying exponents of the form \(a^{m/n}\)?
2. How do you simplify a complex fraction involving roots and exponents?
3. What is the difference between square roots and cube roots in terms of exponentiation?
4. How do you simplify a fraction by factoring the numerator and denominator?
5. Can you apply these exponent rules to expressions with variables?

**Tip:** Remember that fractional exponents represent both roots and powers. For example, \(a^{m/n} = \sqrt[n]{a^m}\).

Simplify: \frac{(25)^{1/2} + (36)^{1/2}}{(125)^{2/3} - (27)^{1/3}}

Simplify the expression \frac{(25)^{1/2} + (36)^{1/2}}{(125)^{2/3} - (27)^{1/3}} by applying rules of exponents and roots. This problem involves the concepts of square roots, cube roots, and fractional exponents, suitable for Grades 9-11.

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