Here are some moderately challenging exponent simplification problems along with step-by-step solutions:

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Problem 1: Simplify $(3x^2y^3)^2 \cdot (x^3y)^4$

Solution:

1. Apply the power rule: $(a^m)^n = a^{m \cdot n}$ $(3x^2y^3)^2 = 3^2x^{2 \cdot 2}y^{3 \cdot 2} = 9x^4y^6$ $(x^3y)^4 = x^{3 \cdot 4}y^{1 \cdot 4} = x^{12}y^4$
2. Combine the results by multiplying: $(9x^4y^6) \cdot (x^{12}y^4) = 9 \cdot x^{4+12} \cdot y^{6+4}$
3. Simplify: $9x^{16}y^{10}$

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Problem 2: Simplify $\frac{x^5y^{-3}}{x^{-2}y^4}$

Solution:

1. Apply the division rule: $\frac{a^m}{a^n} = a^{m-n}$ $\frac{x^5}{x^{-2}} = x^{5 - (-2)} = x^{5+2} = x^7$ $\frac{y^{-3}}{y^4} = y^{-3 - 4} = y^{-7}$
2. Combine the results: $x^7y^{-7}$
3. Simplify further (write with positive exponents): $x^7 \cdot \frac{1}{y^7} = \frac{x^7}{y^7}$

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Problem 3: Simplify $\frac{(2x^3y^{-2})^3}{(4x^2y)^2}$

Solution:

1. Simplify the numerator and denominator separately using $(a^m)^n = a^{m \cdot n}$: $(2x^3y^{-2})^3 = 2^3x^{3 \cdot 3}y^{-2 \cdot 3} = 8x^9y^{-6}$ $(4x^2y)^2 = 4^2x^{2 \cdot 2}y^{1 \cdot 2} = 16x^4y^2$
2. Combine the numerator and denominator: $\frac{8x^9y^{-6}}{16x^4y^2} = \frac{8}{16} \cdot \frac{x^9}{x^4} \cdot \frac{y^{-6}}{y^2}$
3. Simplify each term: $\frac{8}{16} = \frac{1}{2}, \quad \frac{x^9}{x^4} = x^{9-4} = x^5, \quad \frac{y^{-6}}{y^2} = y^{-6-2} = y^{-8}$
4. Final result: $\frac{x^5}{2y^8}$

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Problem 4: Simplify $(a^3b^{-1})^{-2} \cdot (ab)^4$

Solution:

1. Simplify each part:
   • For $(a^3b^{-1})^{-2}$, apply $(a^m)^n = a^{m \cdot n}$: $a^3b^{-1})^{-2} = a^{3 \cdot (-2)}b^{-1 \cdot (-2)} = a^{-6}b^2$
   • For $(ab)^4$, apply $(a^m)^n = a^{m \cdot n}$: $(ab)^4 = a^{1 \cdot 4}b^{1 \cdot 4} = a^4b^4$
2. Multiply the results: $(a^{-6}b^2) \cdot (a^4b^4) = a^{-6+4}b^{2+4}$
3. Simplify: $a^{-2}b^6$
4. Write with positive exponents: $\frac{b^6}{a^2}$

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Problem 5: Simplify $\frac{(3x^{-2}y^4)^2}{9x^4y^5}$

Solution:

1. Simplify the numerator: $(3x^{-2}y^4)^2 = 3^2x^{-2 \cdot 2}y^{4 \cdot 2} = 9x^{-4}y^8$
2. Write the fraction: $\frac{9x^{-4}y^8}{9x^4y^5}$
3. Cancel the constants and apply division rules: $\frac{9}{9} = 1, \quad \frac{x^{-4}}{x^4} = x^{-4-4} = x^{-8}, \quad \frac{y^8}{y^5} = y^{8-5} = y^3$
4. Simplify: $x^{-8}y^3 = \frac{y^3}{x^8}$

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Do you want more of these problems or further clarification on any of the steps? 😊

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5 Related Questions:

1. How do exponent rules apply to negative exponents during division?
2. Can fractional exponents be simplified in a similar way?
3. What happens when zero is involved as an exponent?
4. How do you handle expressions with multiple variables and mixed powers?
5. Why is it important to write answers with positive exponents?

Tip:

Always simplify each part of the expression step by step to avoid mistakes, especially when dealing with negative or fractional exponents!