ANSWER
The two powers are 5 and 3
Step-by-step explanation
Given that;
The product of two powers is 11^8
The quotient of two power is 11^2
Follow the steps below to find the unknown powers
Let the unknown powers be x and y
Step 1; Apply the law of indicies

Since the product of the two powers is 11^8, then we have below equation


Step 2; Solve the two equations simultaneously to find the value of x and y
x + y = 8 ----------- equation 1
x - y = 2 ------------ equation 2
Isolate x in equation 1
x + y = 8
x = 8 - y -------- equation 3
Substitute x = 8- y into equation 2
8 - y - y = 2
8 - 2y = 2
Subtract 8 from both sides of the equation
8 - 8 - 2y = 2 - 8
-2y = -6
Divide both sides by -2
y = -6/-2
y = 3
Find the value of x by substituting y = 3 into equation 1
x + y = 8
x + 3 = 8
x = 8 - 3
x = 5
Therefore, x = 5 and y = 3
Hence, the two powers are 5 and 3