If n is a positive integer, then what is the value of (2/3)^n (1 1/2)^n-1? for positive integer n? Express your answer as a common fraciton?
(2/3)^n (1 1/2)^n-1
=(2/3)^n (3/2)^(n-1)
= (2/3)^n (2/3)^(1 - n)
= (2/3)(n + 1-n)
= (2/3)^1
= 2/3
btw, this is true for all values of n, not just positive integers.
e.g. let n = 2.3
(2/3)^2.3 (3/2)^(2.3-1)
= (2/3)^2.3 (3/2)^(1.3)
= (2/3)^2.3 (2/3)^-1.3
= (2/3)^(2.3 - 1.3)
= (2/3)^1 = 2/3
To find the value of the expression (2/3)^n * (1 1/2)^(n-1) for a positive integer n, we need to simplify it step by step.
Let's first simplify (1 1/2)^(n-1):
1 1/2 can be written as an improper fraction: 3/2. Therefore, we have:
(3/2)^(n-1).
Now, let's simplify (2/3)^n:
To raise a fraction to a power, we raise both the numerator and the denominator to that power. Thus, we have:
(2^n) / (3^n).
Now, we can substitute these simplified expressions back into the original expression:
[(2^n) / (3^n)] * [(3/2)^(n-1)].
Next, let's simplify the second term, (3/2)^(n-1):
(3/2)^(n-1) can be simplified as (3^(n-1)) / (2^(n-1)).
Now, we have:
[(2^n) / (3^n)] * [(3^(n-1)) / (2^(n-1))].
To simplify this expression, we can cancel out common factors in the numerator and denominator.
In the first term, (2^n) / (3^n), there are no common factors to cancel out.
In the second term, (3^(n-1)) / (2^(n-1)), we can cancel out a common factor of 2^(n-1) in the numerator and denominator.
Hence, we are left with:
[(2^n) / (3^n)] * [3^(n-1)].
To multiply these terms, we multiply the numerators together and the denominators together:
(2^n * 3^(n-1)) / (3^n).
Now, we can simplify further:
(2^n) * (3^(n-1)) is equivalent to (2^n) * (3^n/3).
Since we multiplied (2^n) and (3^(n-1)), the exponent of 3 should increase by 1 and we divided by 3 outside the parentheses.
Expanding further, we have:
(2^n * 3^n) / 3.
Finally, to express the answer as a common fraction, we write it as:
(2 * 3^n) / 3.
Therefore, the value of the expression (2/3)^n * (1 1/2)^(n-1) for positive integer n is (2 * 3^n) / 3.