Hi ya!
So I got a question like when I'm rationalzing a denomenator in order to get simple radical form I really don't know what to do
for example this problem
(6^(2^-1)-3^(2^-1))^-1
ok so I know your trying to get it so it's a perfect sqaure right so I can legally write this
(6^(2^-1)-3^(2^-1))^-1 A = (6 - 3)^-1 = 3^-1
or a simplified formula
(6^(2^-1)-3^(2^-1))^-1 A = 3^-1
and solve for A were A equals the value needed to get the perfect square 6-3 or simply 3 right so I solved
A=3^-1(6^(2^-1)-3^(2^-1))
and I get some value which is equal to
(6^(2^-1)+3^(2^-1))^-1
but where s the proof that this is correct using that formula all you get is some number and I have no idea how you get this answer I know you just make the negetive positive but what allows you to do that?????
You still haven't read about conjugates. You are NOT trying to get a perfect square, you are trying to get a difference of perfect squares.
http://www.regentsprep.org/Regents/math/algtrig/ATO3/rdlesson.htm
To simplify and rationalize the denominator, you can follow the steps below:
Step 1: Identify the terms in the denominator that contain radicals.
In the given problem, the terms in the denominator are:
- 6^(2^-1)
- 3^(2^-1)
Step 2: Simplify each term separately.
Let's start with the first term, 6^(2^-1).
To simplify this term, we first need to understand the exponent 2^-1. An exponent of -1 represents taking the reciprocal of the number raised to that power.
So, 2^-1 is equivalent to 1/2^1, which simplifies to 1/2.
Now, the first term becomes:
6^(1/2)
Similarly, for the second term, 3^(2^-1):
2^-1 is equivalent to 1/2^1, which simplifies to 1/2.
So, the second term becomes:
3^(1/2)
Step 3: Combine the simplified terms.
Now that we have simplified each term, we can write the expression as:
(6^(1/2) - 3^(1/2))^-1
Step 4: Rationalize the denominator.
To rationalize the denominator, we multiply both the numerator and the denominator of the fraction by the conjugate of the denominator. The conjugate is obtained by changing the sign between the terms involving radicals.
The conjugate of (6^(1/2) - 3^(1/2)) is (6^(1/2) + 3^(1/2)).
So, by multiplying both numerator and denominator by the conjugate, we get:
[(6^(1/2) - 3^(1/2)) * (6^(1/2) + 3^(1/2))]^-1
Step 5: Simplify further if possible.
Now, we need to multiply the numerator using the distributive property.
(6^(1/2) - 3^(1/2)) * (6^(1/2) + 3^(1/2)) simplifies to:
(6 - 3)
Therefore, the expression becomes:
(6 - 3)^-1
Step 6: Simplify the expression.
(6 - 3)^-1 is equal to 3^-1 since 6 - 3 equals 3.
Thus, the simplified expression is: 3^-1
So, to summarize, the original expression (6^(2^-1) - 3^(2^-1))^-1 simplifies to 3^-1, which is equivalent to 1/3.