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Let's break down the two parts of the problem:

### Problem:
Simplify the following and express the result in the form of a positive exponent.

#### a) \(\left(\frac{3}{7}\right)^2 \times \left(\frac{3}{7}\right)^3\)

To simplify this, we use the rule of exponents:

\[
a^m \times a^n = a^{m+n}
\]

So, we add the exponents:

\[
\left(\frac{3}{7}\right)^2 \times \left(\frac{3}{7}\right)^3 = \left(\frac{3}{7}\right)^{2+3} = \left(\frac{3}{7}\right)^5
\]

Thus, the simplified expression is:

\[
\left(\frac{3}{7}\right)^5
\]

---

#### b) \(\left(\frac{1}{2}\right)^5 + \left(\frac{1}{2}\right)^3\)

We cannot directly combine these two terms because they have different exponents. We can, however, factor out the common base \(\left(\frac{1}{2}\right)^3\):

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

Now, simplify the expression inside the brackets:

\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]

So the expression becomes:

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

Thus, the simplified expression is:

\[
\frac{5}{4} \times \left(\frac{1}{2}\right)^3
\]

---

Would you like further clarifications or additional details?

Here are five questions to explore further:
1. What happens when the base is negative but the exponent is even or odd?
2. How can you simplify expressions with fractional exponents?
3. Can exponents be negative, and how would that affect the simplification process?
4. What are the rules for combining terms with different bases but similar exponents?
5. How do you expand and simplify binomial expressions with exponents?

**Tip:** Always look for common factors when adding terms with exponents to simplify the expression further.

Simplify the following and express the result in the form of a positive exponent: (3/7)^2 × (3/7)^3 and (1/2)^5 + (1/2)^3.

This problem involves simplifying expressions using exponent rules and factoring. Learn how to handle expressions like (3/7)^2 × (3/7)^3 and (1/2)^5 + (1/2)^3. Suitable for students in Grades 8-10.

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