https://r2.erweima.ai/mathbot/image/1726812718771uwxz5741.jpg

The expression in the image is:

\[
\frac{x^{2/3}}{x^{3/2}}
\]

We are tasked with simplifying this and writing the answer in two different forms:

### A) The answer in the form \(x^a\)
To simplify this, apply the laws of exponents:

\[
\frac{x^{2/3}}{x^{3/2}} = x^{2/3 - 3/2}
\]

To subtract the exponents, we need a common denominator for \(2/3\) and \(3/2\):

\[
2/3 = 4/6 \quad \text{and} \quad 3/2 = 9/6
\]

Now subtract the exponents:

\[
\frac{4}{6} - \frac{9}{6} = \frac{-5}{6}
\]

Thus, the simplified expression is:

\[
x^{-5/6}
\]

So, the exponent \(a\) is \(-\frac{5}{6}\).

### B) The answer in the form \(\frac{1}{x^b}\)

We can rewrite \(x^{-5/6}\) in the form \(\frac{1}{x^b}\):

\[
x^{-5/6} = \frac{1}{x^{5/6}}
\]

Thus, the exponent \(b\) is \(\frac{5}{6}\).

---

### Final Answers:
- \(a = -\frac{5}{6}\)
- \(b = \frac{5}{6}\)

Let me know if you'd like more details or explanations.

### Follow-up Questions:
1. What is the general rule for dividing terms with exponents?
2. How can you convert negative exponents into fractions?
3. How would you handle a similar problem with more complex exponents?
4. Why is it necessary to find a common denominator when subtracting fractions?
5. Can this result be verified using logarithmic properties?

**Tip**: Always ensure that you simplify exponents by subtracting when dividing like bases!

Simplify the expression x^(2/3) / x^(3/2). Assume x > 0. Provide the exponent a for the form x^a and the exponent b for the form 1 / x^b.

Learn how to simplify the expression x^(2/3) / x^(3/2) by applying exponent rules. This problem covers key algebraic concepts, including simplifying exponents and converting negative exponents into fractions. Suitable for students in Grades 9-11.

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