Let's evaluate the given expression step by step:

Expression:

$\frac{(-27)^{\frac{2}{3}}}{(4)^{-\frac{5}{2}}}$

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Step 1: Simplify $(-27)^{\frac{2}{3}}$

• The fractional exponent $\frac{2}{3}$ means:
  • Cube root first ($\frac{1}{3}$).
  • Then square the result ($\frac{2}{3}$).

$(-27)^{\frac{1}{3}} = -3 \quad \text{(since the cube root of -27 is -3)}$ $(-3)^2 = 9$ Thus, $(-27)^{\frac{2}{3}} = 9$.

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Step 2: Simplify $(4)^{-\frac{5}{2}}$

• The fractional exponent $-\frac{5}{2}$ means:
  • Take the reciprocal (because of the negative sign).
  • Find the square root ($\frac{1}{2}$).
  • Then raise the result to the power of 5.

$4^{\frac{1}{2}} = \sqrt{4} = 2$ $4^{-\frac{5}{2}} = \frac{1}{2^5} = \frac{1}{32}$

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Step 3: Simplify the division:

$\frac{(-27)^{\frac{2}{3}}}{(4)^{-\frac{5}{2}}} = \frac{9}{\frac{1}{32}}$ Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{9}{\frac{1}{32}} = 9 \cdot 32 = 288$

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Final Answer:

$\boxed{288}$

Let me know if you'd like more clarification! 😊
Here are five related questions for practice:

1. Simplify $(64)^{\frac{2}{3}} \cdot (8)^{-\frac{1}{3}}$.
2. Evaluate $(-16)^{\frac{3}{4}}$.
3. Simplify $(125)^{\frac{4}{3}}$.
4. Solve $\frac{(-8)^{\frac{2}{3}}}{(9)^{-\frac{1}{2}}}$.
5. Expand $(x^3 y^{-2})^{\frac{1}{3}}$.

Tip: For fractional exponents, always apply root and power sequentially to simplify operations effectively.