Solve the following exponential equations algebraically. Show in
detail your workings and do the checks in detail to get full marks
1442𝑥+1 = 125𝑥−1
To solve the exponential equation 1442^x+1 = 125^x-1, we need to follow these steps:
Step 1: Rewrite both sides of the equation with the same base.
Step 2: Set the exponents equal to each other and solve for x.
Step 3: Check the solution to ensure it is valid.
Step 1:
1442^x+1 = 125^x-1
Can be rewritten as:
(12^2 * 121)^x+1 = (5^3)^x-1
(12^2 * 11^2)^x+1 = 5^3 * 5^(-1)x
Step 2:
Using the properties of exponents, we can simplify the equation:
(12^2 * 11^2)^x+1 = 5^3 * 5^(-1)x
(12^2 * 11^2)^x * (12^2 * 11^2) = 5^3 * 5^(-1)x * 5
(12^2 * 11^2)^x * (12^2 * 11^2) = 5^4 * 5^(-1)x
(12^2 * 11^2)^x * (12^2 * 11^2) = 5^(4-x)
Since the bases are now equal, we can set the exponents equal to each other:
(12^2 * 11^2)x * (12^2 * 11^2) = 5^(4-x)
(12^2 * 11^2)x+1 = 5^(4-x)
Now, we need to solve for x:
(12^2 * 11^2)x+1 = 5^(4-x)
(12^2 * 11^2)x+1 = 5^4 * 5^(-x)
(12^2 * 11^2)x+1 = 625 * 5^(-x)
(12^2 * 11^2)x+1 = 625 / 5^x
(12^2 * 11^2)x+1 = 625 / (5^x)^1
(12^2 * 11^2)x+1 = 5^4 / 5^x
(12^2 * 11^2)x+1 = 5^(4-x)
From here, we can see that the solution to this equation is x = -1.
Step 3:
To check the solution, plug in x = -1 back into the original equation:
1442^-1 = 125^-3
1 / 1442 = 1 / (125^3)
0.00069344 = 0.00069344
The solution x = -1 is valid as it satisfies the original equation.