Final answer:
The simplified expression (((6^7/3^3) div (6^6/3^4)^3)) reduces to 2/9 after evaluating exponents, performing division, and canceling out common terms. The value 2/9 is not one of the provided options.
Step-by-step explanation:
Let's simplify the given expression step by step:
(((6⁷/3³) div (6⁶/3⁴)³)
First, we evaluate the exponents and division inside the parentheses:
6⁷ means 6 multiplied by itself 7 times and 3³ means 3 multiplied by itself 3 times.
Similarly, 6⁶ means 6 multiplied by itself 6 times and 3⁴ means 3 multiplied by itself 4 times.
So the expression simplifies to:
(6× 6× 6× 6× 6× 6× 6) / (3× 3× 3)
And after simplifying the denominator of the second fraction:
(6× 6× 6× 6× 6× 6) / (3× 3× 3× 3)
Now, we use the property that (a·b)³ = a³·b³ to expand the cube of the second fraction:
((6⁷/3³) div ((6⁶³)/(3⁴³)))
Which gives:
((6⁷/3³) div (6¹¹/3¹¹) )
Since 6⁷ = 6× 6⁶ and 3¹¹ = 3× 3¹ and the div operator between the fractions signifies division, we can simplify further:
(6×(6⁶/3³)) / (6⁶/3⁴³) )
This simplifies by canceling out the common 6⁶ and 3³ terms leaving:
6 / 3¹ = 6 / 27
And since 6/27 can be simplified to 2/9, the final value of the expression is 2/9, which is not one of the options provided.