) \( 2^{x} \cdot 4^3 \cdot 2^y \)
Write \( 4 \) as \( 2^2 \) and rewrite the given expression as
\( = 2^{x} \cdot (2^2)^3 \cdot 2^y \)
Use rule 8 to rewrite \( (2^2)^3 \) as \( 2^6 \)
\( = 2^{x} \cdot 2^6 \cdot 2^y \)
Use rule 3 extended to three terms with the same base
\( = 2^{x + 6 + y} \)
) \( (3^{-1})^x \)
Use rule 8 to rewrite the given expression as
\( = 3^{(-1) \cdot x} \)
Simplify exponent: \( (-1) \cdot x = - x \)
\( = 3^{-x} \)
Use rule 2 to rewrite as
\( = \dfrac{1}{3^x} \)
) \( \dfrac{y^4 x^3}{x^2 y^2} \)
Rewrite as the product of two fractions where the numerator and denominator of each fraction has the same variable
\( = \dfrac{x^3}{x^2} \cdot \dfrac{y^4}{y^2} \)
Apply rule 5 to each fraction
\( = x^{3-2} \cdot y^{4-2} \)
Simplify
\( = x y^2 \)
) \( \dfrac{x^2}{4y^2} \cdot \left (\dfrac{8y}{x} \right)^2\)
Apply rule 6 to the term \( \left (\dfrac{8y}{x} \right)^2\)
\( = \dfrac{x^2}{4y^2} \cdot \dfrac{(8y)^2}{x^2} \)
Apply rule 4 to the term \( (8y)^2 \)
\( = \dfrac{x^2}{4y^2} \cdot \dfrac{8^2 y^2}{x^2} \)
Rewrite as the product of three fractions
\( = \left(\dfrac{8^2}{4} \cdot \right) \left(\dfrac{x^2}{x^2}\right) \cdot \left(\dfrac{y^2}{y^2}\right) \)
Rewrite \( 8^2 \) as \( (2^3)^2 = 2^6 \) and \( 4 \) as \( 2^2 \)
\( = \left(\dfrac{2^6}{2^2} \cdot \right) \left(\dfrac{x^2}{x^2}\right) \cdot \left(\dfrac{y^2}{y^2}\right) \)
Use rule 5 to the fraction \( \dfrac{2^6}{2^2} \) and cancel same terms in the other two fractions
\( = 2^{6-2} \cdot 1 \cdot 1 = 2^4 = 16 \)
) \( ( - 6 a)^2 \cdot (a^2 + 1)^0 \)
Use rule 9 to simplify \( (a^2 + 1)^0 \) to 1
\( = ( - 6 a)^2 \cdot 1 \)
Use rule 4 extended to the three terms in \( ( - 6 a)^2 \)
\( = (-1)^2 6^2 a^2 \)
Simplify
\( = 36 a^2 \)