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, $a^{m/n} = \sqrt[n]{a^m}$.