(1/2)^3(2/3)^4/5/6^3 Show work

To simplify the given expression (1/2)^3 × (2/3)^4 / (5/6)^3, we can use the rules of exponentiation and fraction arithmetic.

First, let's simplify each term separately:

1. Simplifying (1/2)^3:
To raise a fraction to a power, we raise both the numerator and denominator to that power.
(1/2)^3 = (1^3) / (2^3) = 1/8

2. Simplifying (2/3)^4:
Similarly, we raise both the numerator and denominator to the power.
(2/3)^4 = (2^4) / (3^4) = 16/81

3. Simplifying (5/6)^3:
Again, we raise both the numerator and denominator to the power.
(5/6)^3 = (5^3) / (6^3) = 125/216

Now that we have simplified each term, let's substitute these values back into the original expression:

1/8 × 16/81 / 125/216

To divide fractions, we multiply by the reciprocal of the divisor. So, we can rewrite the division as multiplication by the reciprocal:

1/8 × 16/81 × 216/125

Next, we multiply the numerators together and the denominators together:

(1 × 16 × 216) / (8 × 81 × 125)

Now, we can cancel out common factors between the numerator and denominator:

(1 × 16 × 3 × 6) / (1 × 9 × 125)

Simplifying further:

(16 × 3 × 6) / (125 × 9)

Finally, we multiply the remaining numbers together:

288 / 1125

Therefore, the simplified value of the expression (1/2)^3 × (2/3)^4 / (5/6)^3 is 288/1125.