could you please tell me how to turn this into simple radical form
2(3^(2^-1)-1)^-1
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With this kind of question, we only need to do the simplification methodically.
Start by checking all the parentheses to see if they are balanced or missing.
In the expression given, the parentheses are balanced, so that's a good start.
The -1 following the exponent operator (^) should preferably have parentheses around it to avoid confusion or ambiguity. So I am changing the expression to:
2(3^(2^-1)-1)^(-1)
If this is not the case, let me know.
From the study of exponents, we understand that x^(-1) is the same as 1/x. So we will start by removing the last exponent:
2(3^(2^-1)-1)^(-1)
=2/(3^(2^-1)-1)
The same goes for the exponent of -1 for 2, which makes
2/(3^(2^-1)-1)
=2/(3^(1/2)-1)
We have also learned that
x^(1/2) = sqrt(x), so the expression can further be reduced to:
2/(3^(1/2)-1)
=2/(sqrt(3)-1)
In general, we do not like to have expressions involving square-roots dangling in the denominator, if at all possible. This can be achived by multiplying both the numerator and denominator by the expression (sqrt(3)+1), which reduces the numerator to 2(sqrt(3)+1), and the numerator to
(sqrt(3)^2-1^2)
=3-1
=2
Cancelling the common factor 2, we should obtain the final answer as
sqrt(3)+1
Note: I have simplified the expression as I type, so I ask you to do a good job of checking my work and correct mistakes if there are any.
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Please start a new thread for your DVD question. I prefer to know if this question came from Jen or someone else.
To turn the expression 2(3^(2^-1)-1)^-1 into simple radical form, we can follow these steps:
Step 1: Simplify the exponent
In this case, the exponent 2^-1 can be simplified to its reciprocal, which is 1/2.
Step 2: Substitute the simplified exponent
Now, we replace the original expression with the simplified exponent:
2(3^(1/2)-1)^-1
Step 3: Simplify the innermost parentheses
Inside the parentheses, we have sqrt(3) - 1.
Step 4: Apply the exponent outside the parentheses
Since the exponent is -1, we need to take the reciprocal of the expression inside the parentheses:
2 / (sqrt(3) - 1)
Step 5: Rationalize the denominator
To rationalize the denominator (remove the square root in the denominator), we multiply the numerator and denominator by the conjugate of the denominator, which is (sqrt(3) + 1):
2 / (sqrt(3) - 1) * (sqrt(3) + 1) / (sqrt(3) + 1)
Step 6: Simplify the denominator
By using the difference of squares identity, we can expand the denominator:
2 / [(sqrt(3))^2 - (1)^2]
= 2 / (3 - 1)
= 2 / 2
= 1
So, the expression 2(3^(2^-1)-1)^-1 simplifies to 1 in simple radical form.