Answer:
Step 1: Simplify the exponents on both sides of the equation. Recall that (a^b)^c = a^(b*c), so we can write:
(1/216)^(3v-1) = (1/6)^(2v-2)
= (6^-1)^(2v-2) (since 1/6 = 6^-1)
= 6^(-(2v-2)) (since (a^-b) = 1/(a^b))
= 6^(2-2v) (since -(2v-2) = -2v + 2)
Step 2: Rewrite 1/216 as a power of 6. We have:
1/216 = 6^(-3)
Substituting this into our equation gives:
6^(-3(3v-1)) = 6^(2-2v)
Step 3: Use the property that if a^x = a^y, then x = y. Since both sides of the equation have the same base (6), we can equate the exponents:
-3(3v-1) = 2-2v
Step 4: Solve for v. First, simplify the left-hand side of the equation:
-3(3v-1) = -9v + 3
Now we can rewrite the equation as:
-9v + 3 = 2 - 2v
Step 5: Solve for v. Add 9v to both sides and subtract 2 from both sides:
-7v = -1
Finally, divide both sides by -7 to obtain:
v = 1/7
Therefore, the solution to the equation is v = 1/7.
Verification
To verify that v = 1/7 is the solution to the equation:
(1/216)^(3v-1) = (1/6)^(2v-2)
We can substitute v = 1/7 into the equation and simplify both sides to see if they are equal:
Left-hand side:
(1/216)^(3v-1) = (1/216)^(3(1/7)-1) = (1/216)^(2/7)
Right-hand side:
(1/6)^(2v-2) = (1/6)^(2(1/7)-2) = (1/6)^(-10/7) = (6/1)^10/7
Simplifying:
(1/216)^(2/7) = (6/1)^(10/7)
Taking the seventh root of both sides:
1/6 = 1/6
Since both sides of the equation are equal, we have verified that v = 1/7 is the solution to the equation.