Calculate the result of the expression 3/10 x 5/6 x 1/9 x 3/4 and simplify to lowest terms.

To calculate the expression ( \frac{3}{10} \times \frac{5}{6} \times \frac{1}{9} \times \frac{3}{4} ) and simplify it to its lowest terms, follow these steps:

First, multiply the numerators together and the denominators together:

Numerators: ( 3 \times 5 \times 1 \times 3 = 45 )

Denominators: ( 10 \times 6 \times 9 \times 4 = 2160 )

Now, combine these into a single fraction:

[

\frac{45}{2160}

]

Next, simplify this fraction by finding the greatest common divisor (GCD) of 45 and 2160.

The prime factors of 45 are ( 3^2 \times 5 ) and the prime factors of 2160 (which can be calculated by dividing by prime numbers) yield ( 2^4 \times 3^3 \times 5 ).

The GCD in this case is ( 15 ) (the highest power of common primes), as both the numerator and denominator share the factors of ( 3^1 \