Answer:
144
Explanation:
Given: a1 = 729 , a7 = 64 , a5 = ?
The explicit formula for the term an of a geometric sequence that has the ratio r is : an = a1 *r^(n-1)
The sequence is 729, 729*r, 729* r², 729*r³,....
a7= 729* r^6
64 = 729 * r^6, divide both sides by 729
64/729 = r^6, take the root or raze to the (1/6) th power both sides to find r
(64/729) ^ (1/6) = r
a5 = 729* r^4 , using the explicit formula
a5 = 729* ((64/729) ^(1/6) )^4, substitute the ratio we found
a5 = 729* (64/729) ^(4/6) , use the fact that (a^b)^c = a^(b*c)
a5 = 729* (64/729) ^(2/3), simplify the fraction 4/6 as 2/3
a5 = 729* ((64 ^(2/3) )/ 729^(2/3)), raise a fraction to power (a/b)^c = (a^c)/(b^c)
a5 = 729* (∛64²/∛729²), use rule a^(b/c) = square root base c of a^b
a5 = 729* (16/81), use calculator
a5 = 11,664/ 81, use rule a(b/c) = (a*b)/ c
a5 = 144