Answer
$\frac{x^{18} y^{6}}{4}$
Work Step by Step
$\frac{{({2^{-1}}{x^{-2} {y^{-1}}})}^{-2}{({2}{x^{-4} {y^{3}}})}^{-2}{({16}{x^{-3} {y^{3}}})}^{0}}{(2 x^{-3} y ^ {-5})^2}$
When a product is raised to an exponent, raise each factor to that exponent. Remember, any non-zero base raised to the power of zero equals one.
= $ \frac{(2^{ (-1 \times -2)} \times x^{ (-2 \times -2)} \times y^{ (-1 \times -2)}) \times (2^{ ( -2)} \times x^{ (-4 \times -2)} \times y^{ (3 \times -2)}) \times (16^{ (0)} \times x^{ (-3 \times 0)} \times y^{ (3\times 0)})}{(2^{ 1\times 2)} \times x^{ (-3 \times 2)} \times y^{ (-5\times 2)})}$
= $\frac{( 2^{2} \times x^{4} \times y^{2} ) \times( 2^{-2} \times x^{8} \times y^{-6} ) \times 1}{( 2^{2} \times x^{-6} \times y^{-10} )}$
Group like terms.
= $( \frac{ 2^{2} \times 2^{-2}}{2^{2}})( \frac{ x^{4} \times x^{8}}{x^{-6}})( \frac{ y^{2} \times y^{-6}}{y^{-10}})$
When multiplying exponential expression with the same non-zero base add the exponents. Use this sum as the exponent of the common base.
= $( \frac{ 2^{2 + (-2) } }{2^{2}})( \frac{ x^{(4 + 8)} }{x^{-6}})( \frac{ y^{(2 +(-6))} }{y^{-10}})$
= $( \frac{ 2^{0} }{2^{2}})( \frac{ x^{12)} }{x^{-6}})( \frac{ y^{-4)} }{y^{-10}})$
When dividing exponential expressions with the same non-zero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.
= $ 2^{(0-2)} \times x^{(12-(-6))} \times y^{(-4-(-10))} $
= $ 2^{(-2)} \times x^{(12+6)} \times y^{(-4+10)} $
= $ 2^{-2} \times x^{18} \times y^{6} $
= $ 2^{-2} x^{18} y^{6} $
When an exponent is negative, write the expression as a fraction and move the base from the numerator to the denominator (or vise-versa) and make the exponent positive.
= $\frac{{x^{18} y^{6}}}{2^{2}}$
= $\frac{{x^{18} y^{6}}} {4}$
