https://file.aiquickdraw.com/mathbot/file/1726744173233vc13z01j.jpg

Let's solve the exercises one by one from the image.

### Instructions: Convert the given expressions into a single exponent form.

#### a) \( 3^{\frac{1}{3}} \cdot 3^{\frac{5}{2}} \)

- Use the rule for multiplying powers of the same base: \( a^m \cdot a^n = a^{m+n} \).
\[
3^{\frac{1}{3}} \cdot 3^{\frac{5}{2}} = 3^{\left(\frac{1}{3} + \frac{5}{2}\right)}
\]
Now, find the common denominator:
\[
\frac{1}{3} = \frac{2}{6}, \quad \frac{5}{2} = \frac{15}{6}
\]
Add the exponents:
\[
\frac{2}{6} + \frac{15}{6} = \frac{17}{6}
\]
Thus, the final result is:
\[
3^{\frac{17}{6}}
\]

#### b) \( 2^{\frac{1}{2}} \cdot 2^{\frac{-1}{3}} \)

- Again, apply the multiplication rule for exponents:
\[
2^{\frac{1}{2}} \cdot 2^{\frac{-1}{3}} = 2^{\left(\frac{1}{2} + \frac{-1}{3}\right)}
\]
Find the common denominator:
\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{-1}{3} = \frac{-2}{6}
\]
Add the exponents:
\[
\frac{3}{6} + \frac{-2}{6} = \frac{1}{6}
\]
Thus, the result is:
\[
2^{\frac{1}{6}}
\]

#### c) \( 5^{\frac{1}{2}} \cdot 5^{\frac{1}{3}} \)

- Apply the same rule:
\[
5^{\frac{1}{2}} \cdot 5^{\frac{1}{3}} = 5^{\left(\frac{1}{2} + \frac{1}{3}\right)}
\]
Find the common denominator:
\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}
\]
Add the exponents:
\[
\frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\]
Thus, the result is:
\[
5^{\frac{5}{6}}
\]

#### d) \( \frac{3^{\frac{7}{3}}}{3^{\frac{1}{3}}} \)

- Use the division rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
\[
\frac{3^{\frac{7}{3}}}{3^{\frac{1}{3}}} = 3^{\left(\frac{7}{3} - \frac{1}{3}\right)}
\]
Subtract the exponents:
\[
\frac{7}{3} - \frac{1}{3} = \frac{6}{3} = 2
\]
Thus, the result is:
\[
3^2
\]

#### e) \( \frac{7^{\frac{7}{3}}}{7^{\frac{-2}{3}}} \)

- Apply the same division rule:
\[
\frac{7^{\frac{7}{3}}}{7^{\frac{-2}{3}}} = 7^{\left(\frac{7}{3} - \frac{-2}{3}\right)}
\]
Subtract the exponents:
\[
\frac{7}{3} - \frac{-2}{3} = \frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3
\]
Thus, the result is:
\[
7^3
\]

#### f) \( \frac{7^{\frac{1}{2}}}{7^{\frac{1}{3}}} \)

- Apply the division rule:
\[
\frac{7^{\frac{1}{2}}}{7^{\frac{1}{3}}} = 7^{\left(\frac{1}{2} - \frac{1}{3}\right)}
\]
Find the common denominator:
\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}
\]
Subtract the exponents:
\[
\frac{3}{6} - \frac{2}{6} = \frac{1}{6}
\]
Thus, the result is:
\[
7^{\frac{1}{6}}
\]

---

Would you like further clarification or additional details on any of these steps?

### Related Questions:
1. How do you simplify expressions with negative exponents?
2. Can you explain the power rule for fractional exponents in more detail?
3. What are some common mistakes to avoid when adding exponents?
4. How would you approach exponents when the bases are different?
5. Why do we need a common denominator when adding or subtracting fractional exponents?

### Tip:
When dealing with fractional exponents, always ensure the fractions are simplified before adding or subtracting them to avoid unnecessary complexity.

Tulis setiap bentuk berikut sebagai satu bilangan berpangkat.

Solve exponent expressions by applying exponent rules for multiplication and division. This problem involves adding and subtracting fractional exponents. Ideal for students in Grades 8-10.

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