To simplify the given expression, let us carefully apply the laws of indices step by step. Here's the expression:

$\frac{(c^{3/2} h^{-1} b^2 k^1)^2 (k^{-3/2} b^1)^1}{(h^1 b^1 c^{3/2})^2}$

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Step 1: Expand each term using the laws of indices

1. Expand $(c^{3/2} h^{-1} b^2 k^1)^2$:

   • Each exponent is multiplied by $2$: $c^{3/2 \cdot 2} h^{-1 \cdot 2} b^{2 \cdot 2} k^{1 \cdot 2} = c^3 h^{-2} b^4 k^2$

2. Expand $(k^{-3/2} b^1)^1$:

   • Raising to the power of $1$ leaves the expression unchanged: $k^{-3/2} b^1$

3. Expand $(h^1 b^1 c^{3/2})^2$:

   • Each exponent is multiplied by $2$: $h^{1 \cdot 2} b^{1 \cdot 2} c^{3/2 \cdot 2} = h^2 b^2 c^3$

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Step 2: Combine the terms in the numerator

The numerator is: $(c^3 h^{-2} b^4 k^2) (k^{-3/2} b^1)$

• Combine the powers of $c$, $h$, $b$, and $k$ using the product rule ($a^m \cdot a^n = a^{m+n}$):
  • $c$: $c^3$
  • $h$: $h^{-2}$
  • $b$: $b^{4+1} = b^5$
  • $k$: $k^{2 + (-3/2)} = k^{4/2 - 3/2} = k^{1/2}$

Thus, the numerator simplifies to: $c^3 h^{-2} b^5 k^{1/2}$

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Step 3: Simplify the denominator

The denominator is: $h^2 b^2 c^3$

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Step 4: Divide the numerator by the denominator

Using the quotient rule ($a^m / a^n = a^{m-n}$):

1. $c$: $c^{3-3} = c^0 = 1$ (cancels out completely)
2. $h$: $h^{-2 - 2} = h^{-4}$
3. $b$: $b^{5-2} = b^3$
4. $k$: $k^{1/2}$ (no $k$ in the denominator, so it remains $k^{1/2}$)

The final simplified expression is: $b^3 h^{-4} k^{1/2}$

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Step 5: Match the result with the options

This matches: $\boxed{\text{b3h-4k1/2}}$

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Do you want further clarifications or detailed explanation?

Here are 5 related questions for practice:

1. How do you handle negative exponents when simplifying fractions?
2. Why does $c^0 = 1$ when simplifying expressions?
3. How can you simplify expressions involving multiple variables?
4. What are the key differences between product and quotient rules for exponents?
5. How do fractional exponents (e.g., $k^{1/2}$) relate to roots?

Tip: Always combine terms systematically, keeping track of the variables one by one to avoid mistakes.