Question

1. \\(\frac{4a^{3}b^{4}}{4a^{-4}b^{-3} \cdot 2a^{-4}b^{2}}\\)

Explanation:

Step1: Simplify denominator constants

Multiply the constant terms:
$4 \cdot 2 = 8$

Step2: Combine $a$-terms in denominator

Add exponents of like bases:
$a^{-4} \cdot a^{-4} = a^{-4 + (-4)} = a^{-8}$

Step3: Combine $b$-terms in denominator

Add exponents of like bases:
$b^{-3} \cdot b^{2} = b^{-3 + 2} = b^{-1}$

Step4: Rewrite the entire expression

Substitute simplified denominator into the fraction:
$\frac{4a^{3}b^{4}}{8a^{-8}b^{-1}}$

Step5: Simplify constants

Reduce the constant fraction:
$\frac{4}{8} = \frac{1}{2}$

Step6: Simplify $a$-terms

Subtract denominator exponent from numerator exponent:
$a^{3 - (-8)} = a^{3 + 8} = a^{11}$

Step7: Simplify $b$-terms

Subtract denominator exponent from numerator exponent:
$b^{4 - (-1)} = b^{4 + 1} = b^{5}$

Step8: Combine all simplified parts

Multiply the constant and variable terms together.

Answer:

$\frac{1}{2}a^{11}b^{5}$