Answer:
\(\dfrac{z-t}{z-y}\)
Step-by-step explanation:
Given multiplication:
\(\dfrac{3(x+y)^{10}}{18(z-t)^{19}} \times \dfrac{10(x+y)(z-t)^{4}}{12(x+y)^{3}(z-y)} \times \dfrac{108(x+y)^{2}(z-t)^{20}}{15(z-t)^{4}(x+y)^{10}}\)
When we multiply fractions, the general rule is to multiply the numerators together and the denominators together. Therefore:
\(\dfrac{3\times 10 \times 108(x+y)^{10}(x+y)(z-t)^{4}(x+y)^{2}(z-t)^{20}}{18\times 12 \times 15(z-t)^{19}(x+y)^{3}(z-y)(z-t)^{4}(x+y)^{10}}\)
Multiply the numbers:
\(\dfrac{3240(x+y)^{10}(x+y)(z-t)^{4}(x+y)^{2}(z-t)^{20}}{3240(z-t)^{19}(x+y)^{3}(z-y)(z-t)^{4}(x+y)^{10}}\)
Cancel the common factors 3240, (x + y)¹⁰ and (z - t)⁴:
\(\dfrac{(x+y)(x+y)^{2}(z-t)^{20}}{(z-t)^{19}(x+y)^{3}(z-y)}\)
Apply the product rule of exponents to the numerator, which states that when multiplying two expressions with the same base, we add the exponents. Therefore:
\(\dfrac{(x+y)^{1+2}(z-t)^{20}}{(z-t)^{19}(x+y)^{3}(z-y)} \\\\\\\\ \dfrac{(x+y)^{3}(z-t)^{20}}{(z-t)^{19}(x+y)^{3}(z-y)}\)
Cancel the common factor (x + y)³:
\(\dfrac{(z-t)^{20}}{(z-t)^{19}(z-y)}\)
Now, apply the quotient rule of exponents, which states that when dividing two expressions with the same base, we subtract the exponents:
\(\dfrac{(z-t)^{20-19}}{(z-y)} \\\\\\\\ \dfrac{(z-t)^1}{(z-y)} \\\\\\\\ \dfrac{z-t}{z-y}\)
Therefore, the given expression simplified is:
\(\Large\boxed{\boxed{\dfrac{z-t}{z-y}}}\)
