To simplify the expression \(\left(\frac{25}{9x}\right)^{-\frac{3}{2}}\) using surd form, follow these steps:

1. **Rewrite the expression**:
   \[
   \left(\frac{25}{9x}\right)^{-\frac{3}{2}} = \frac{1}{\left(\frac{25}{9x}\right)^{\frac{3}{2}}}
   \]
   Since the exponent is negative, the expression becomes its reciprocal.

2. **Simplify the inner term**:
   The expression \(\left(\frac{25}{9x}\right)^{\frac{3}{2}}\) can be split as:
   \[
   \left(\frac{25}{9x}\right)^{\frac{3}{2}} = \left(\frac{25}{9}\right)^{\frac{3}{2}} \times (x)^{-\frac{3}{2}}
   \]

3. **Simplify the fraction**:
   First, simplify \(\left(\frac{25}{9}\right)^{\frac{3}{2}}\):
   \[
   \left(\frac{25}{9}\right)^{\frac{3}{2}} = \frac{25^{\frac{3}{2}}}{9^{\frac{3}{2}}}
   \]
   \[
   25^{\frac{3}{2}} = (5^2)^{\frac{3}{2}} = 5^3 = 125
   \]
   \[
   9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}} = 3^3 = 27
   \]
   So:
   \[
   \left(\frac{25}{9}\right)^{\frac{3}{2}} = \frac{125}{27}
   \]

4. **Substitute back**:
   Now, the expression becomes:
   \[
   \frac{1}{\frac{125}{27} \cdot x^{-\frac{3}{2}}}
   \]
   This simplifies to:
   \[
   \frac{27}{125} \cdot x^{\frac{3}{2}}
   \]

5. **Express in surd form**:
   The exponent \(\frac{3}{2}\) means we can write \(x^{\frac{3}{2}}\) as:
   \[
   x^{\frac{3}{2}} = x \sqrt{x}
   \]

So the final simplified expression in surd form is:
\[
\frac{27x\sqrt{x}}{125}
\]

Would you like further details on any step?

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Here are 5 related questions for practice:

1. Simplify \(\left(\frac{16}{25x^2}\right)^{-\frac{3}{2}}\) in surd form.
2. Simplify \(\left(\frac{49}{64x^4}\right)^{-\frac{1}{2}}\) using surds.
3. Simplify \(\left(\frac{36x}{81}\right)^{\frac{3}{2}}\) in surd form.
4. Simplify \( \left( \frac{8}{x^2} \right)^{-\frac{2}{3}} \) in surd form.
5. Simplify \( \left( \frac{x^2}{4} \right)^{\frac{5}{2}} \) using surds.

**Tip**: When simplifying with fractional exponents, always consider expressing them as radicals (surds) to make interpretation easier.

Learn how to simplify the expression (25/9x)^-3/2 using surd form. This detailed step-by-step solution covers key algebraic concepts such as exponents, rational exponents, and surds. Suitable for students in Grades 10-12.

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