Step-by-step answer:
Given:
(1/7)^(3a+3) = 343^(a-1) ................................................(1)
Find the value of a that satisfies equation (1)
Solution:
Using the law of exponents, (1/x)^p = (x)^(-p), and apply to LHS (left-hand side) of (1)
(7)^(-3a-3) = 343^(a-1) ...................................................(2)
We know that 7^3 = 343, substitute on the RHS (right-hand side) of (2)
(7)^(-3a-3) = (7^3)^(a-1) ...................................................(3)
Using the law of exponents, (x^p)^q = x^(pq) and apply to the RHS of (3)
(7)^(-3a-3) = (7)^(3(a-1)) = 7^(3a-3) ...................................................(3)
now that the bases on each side are equal, we apply the rule
x^(p) = x^(q) => p=q for all x>0, and apply to (3)
7^(-3a-3) = 7(3a-3) =>
-3a-3 = 3a-3............................................................................................(4)
Solve (4) for a
-3 + 3 = 3a+3a
0 = 6a
a=0..........................................................................................................(5)
Check:
Substitute a=0 in (1)
LHS : (1/7)^(0+3) = (1/7)^(3) = (7)^(-3)
RHS : 343^(0-1) = (7^3)^(-1) = (7)^(-3)
Since LHS equals RHS, the solution is validated.
