I would like to know if the following are correct and if not how and what did I do wrong.
27^-2/3
27^-1/3 = 27^-2/3 - -1/3 = 27^-3/3 sqrt 27
(a^1/2b)^1/2 (ab^1/2) = a^5/4b
since 27 = 3^3,
27^(1/3) = 3
27^(2/3) = 3^2 = 9
So, you have
27^-(2/3) = 1/27^(2/3) = 1/9
the other is correct
Thank you Steve
To determine if the expressions are correct or not, let's break down each one and simplify them:
1. 27^-2/3:
To simplify this expression, we can use exponent rules. When a negative exponent is present, we can rewrite it as the reciprocal with a positive exponent. So, 27^-2/3 can be written as 1/27^2/3. Now, let's simplify 27^(2/3).
Since 27 can be written as 3^3, we have:
1/27^(2/3) = 1/(3^3)^(2/3) = 1/3^(3 * 2/3) = 1/3^2 = 1/9.
So, the correct simplified expression is 1/9.
2. 27^-1/3:
Similar to the previous expression, we can rewrite this as 1/27^(1/3). Simplifying the exponent 27^(1/3):
27^(1/3) = (3^3)^(1/3) = 3^(3 * 1/3) = 3^1 = 3.
Therefore, the simplified expression is 1/3.
It seems there was a confusion in the provided solution: 27^-1/3 is not equal to 27^-2/3. The correct answer is 1/3, not 27^-2/3.
Moving on to the second expression:
3. (a^1/2b)^1/2 (ab^1/2):
To simplify this expression, we can distribute the exponent of 1/2 to both terms inside the parentheses:
(a^1/2b)^1/2 = a^(1/2 * 1/2) * b^(1/2 * 1/2) = a^(1/4) * b^(1/4).
Now, let's multiply this simplified expression with the second term, ab^(1/2):
(a^1/4 * b^1/4) * (ab^1/2) = a^(1/4) * b^(1/4) * a * b^(1/2).
To simplify further, we can combine the exponents with the same base:
a^(1/4) * a = a^(1/4 + 1) = a^(5/4).
b^(1/4) * b^(1/2) = b^(1/4 + 1/2) = b^(3/4).
So, the final simplified expression is a^(5/4) * b^(3/4), not a^(5/4b).
In summary, the correct simplified expressions are:
1. 27^-2/3 = 1/9
2. 27^-1/3 = 1/3
3. (a^1/2b)^1/2 (ab^1/2) = a^(5/4) * b^(3/4).