(5^1/2*5^-3/2)^-1/4
5^(1/6) is the sixth root of 5. This means that if we multiply any number by itself 6 times and get 5, then that number is 5^(1/6). Since 5^(1/6) can't be simplified further, we can approximate it using a calculator.
The approximate value of 5^(1/6) is 1.430969... (rounded to 6 decimal places).
AAAaannndd the bot gets it wrong yet again!
(5^(1/2) * 5^(-3/2))^(-1/4)
= (5^-1)^(-1/4)
= 5^(1/4)
Apologies for the mistake in my previous response. You are correct, the expression can be simplified as follows:
(5^(1/2) * 5^(-3/2))^(-1/4)
= (5^(-1))^(-1/4) (using the rule (a^m * a^n) = a^(m+n))
= 5^(1/4)
So, the correct answer is 5^(1/4). Thank you for bringing this to my attention.
To simplify this expression, we can follow the order of operations: parentheses, exponents, multiplication/division (from left to right), and then addition/subtraction (from left to right).
Let's break the expression down step by step:
Step 1: Simplifying the exponents
- The exponent on 5^1/2 means taking the square root of 5.
- √5 = 2.236 (approximately)
- The exponent on 5^-3/2 means taking the square root and then taking the reciprocal of 5^3.
- (5^3)^-1/2 = 1/√(5^3) = 1/(√125) = 1/11.180 (approximately)
Now, let's substitute these values back into the original expression:
Step 2: Substitute the simplified exponents
- (5^1/2 * 5^-3/2)^-1/4
= (2.236 * 1/11.180)^-1/4
Step 3: Simplify the multiplication
- 2.236 * 1/11.180 = 0.2
Step 4: Simplify the exponent
- (0.2)^-1/4 = 1/√(0.2) = 1/0.447 = 2.236 (approx)
So, the simplified value of (5^1/2 * 5^-3/2)^-1/4 is approximately 2.236.
We can simplify the expression as follows:
(5^(1/2) * 5^(-3/2))^(-1/4)
= 5^((1/2) * (-1/4)) * 5^((-3/2) * (-1/4)) (using the rule (a^m * a^n) = a^(m+n))
= 5^(-1/8) * 5^(3/8)
= 5^(3/8 - 1/8)
= 5^(1/2)
So the answer is √5 or 5^(1/2).