Question

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1. $\frac{a^{5}b^{6} cdot a^{6}b^{3}}{(a^{4}b^{-5})^{5}}$

Explanation:

Step1: Simplify numerator (product rule)

When multiplying terms with the same base, add exponents:
$a^5 \cdot a^6 = a^{5+6} = a^{11}$, $b^6 \cdot b^3 = b^{6+3} = b^9$
Numerator becomes: $a^{11}b^{9}$

Step2: Simplify denominator (power rule)

When raising a power to a power, multiply exponents:
$(a^4)^5 = a^{4 \times 5} = a^{20}$, $(b^{-5})^5 = b^{-5 \times 5} = b^{-25}$
Denominator becomes: $a^{20}b^{-25}$

Step3: Divide terms (quotient rule)

When dividing terms with the same base, subtract exponents:
For $a$: $a^{11-20} = a^{-9}$
For $b$: $b^{9 - (-25)} = b^{9+25} = b^{34}$
Result: $a^{-9}b^{34}$

Step4: Rewrite positive exponents

$a^{-9} = \frac{1}{a^9}$, so combine terms:
$\frac{b^{34}}{a^9}$

Answer:

$\frac{b^{34}}{a^9}$