To simplify the given expression in bases 2 and 3, we'll rewrite the powers of \(y\) using bases 2 and 3, then simplify the expression accordingly.

The original expression is:

\[
\frac{y^{3/2} - y^{5/2} - y^{7/3}}{y^{1/2} - y^{3/2} - y^{7/6}}
\]

### Step 1: Rewrite each exponent in terms of logarithms using bases 2 and 3

Let:
\[
y = 2^a \quad \text{and} \quad y = 3^b
\]

This implies:
\[
y^{m/n} = (2^a)^{m/n} = 2^{(a \cdot m/n)} \quad \text{and} \quad y^{m/n} = (3^b)^{m/n} = 3^{(b \cdot m/n)}
\]

So, let's rewrite the terms:

For base 2:
\[
y^{3/2} = 2^{3a/2}, \quad y^{5/2} = 2^{5a/2}, \quad y^{7/3} = 2^{7a/3}
\]

For base 3:
\[
y^{3/2} = 3^{3b/2}, \quad y^{5/2} = 3^{5b/2}, \quad y^{7/3} = 3^{7b/3}
\]

### Step 2: Substitute these into the expression

For base 2:
\[
\frac{2^{3a/2} - 2^{5a/2} - 2^{7a/3}}{2^{a/2} - 2^{3a/2} - 2^{7a/6}}
\]

For base 3:
\[
\frac{3^{3b/2} - 3^{5b/2} - 3^{7b/3}}{3^{b/2} - 3^{3b/2} - 3^{7b/6}}
\]

### Step 3: Simplify by factoring common terms

For base 2, factor out the smallest power of 2:
\[
\frac{2^{a/2}(2^{a} - 2^{2a} - 2^{7a/6})}{2^{a/2}(1 - 2^{a} - 2^{5a/6})}
\]

Cancel \(2^{a/2}\) from both the numerator and the denominator:
\[
\frac{2^{a} - 2^{2a} - 2^{7a/6}}{1 - 2^{a} - 2^{5a/6}}
\]

For base 3, factor out the smallest power of 3:
\[
\frac{3^{b/2}(3^{b} - 3^{2b} - 3^{7b/6})}{3^{b/2}(1 - 3^{b} - 3^{5b/6})}
\]

Cancel \(3^{b/2}\) from both the numerator and the denominator:
\[
\frac{3^{b} - 3^{2b} - 3^{7b/6}}{1 - 3^{b} - 3^{5b/6}}
\]

### Final Simplified Expressions

- **In base 2:** 
\[
\frac{2^{a} - 2^{2a} - 2^{7a/6}}{1 - 2^{a} - 2^{5a/6}}
\]

- **In base 3:**
\[
\frac{3^{b} - 3^{2b} - 3^{7b/6}}{1 - 3^{b} - 3^{5b/6}}
\]

These are the simplified forms of the given expression in bases 2 and 3, respectively.

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**Related Questions:**

1. What is the significance of using different bases in exponents?
2. How does changing the base of the logarithm affect the simplification of expressions?
3. How can you generalize the simplification process for any base?
4. What are common logarithms, and how do they relate to bases 2 and 3?
5. How does factoring work when dealing with exponents of different bases?
6. What would happen if you chose another base like 5 or 10?
7. Can the expression be simplified further by combining like terms in the exponent?
8. How does the simplification process change when working with logarithms in natural bases (e.g., base \(e\))?

**Tip:** When simplifying expressions, consider factoring out the smallest base or exponent to reduce complexity before further manipulation.

Learn how to simplify the given exponential expression in bases 2 and 3 using logarithms and exponent rules. This problem involves converting exponents to logarithms, factoring common terms, and simplifying algebraic expressions. Suitable for advanced high school students studying exponentials and logarithms.

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