Dividing square roots by cube roots

[Angierozm]

I need help the the problem:
27^1/2/9^1/3
First, I multiplied 27^1/2 by 27^3/3 and 9^1/3 by 9^2/2
Then, I multiplied the resulting numbers: 27^3/6/9^2/6 x 9^4/6/9^4/6
which gave me: (27^3/6 x 9^4/6)/9
which I then simplified to (27³ x 9⁴)^1/6/9
Which I finally simplified to 4374/9=483

My problem is that 483 is too big. If i put 27^1/2/9^1/3 into the calculator, I get 2.498.
What did I do wrong?

 

[Bob Brown MSEE]

I need help the the problem:
27^1/2/9^1/3
First, I multiplied 27^1/2 by 27^3/3 and 9^1/3 by 9^2/2
Then, I multiplied the resulting numbers: 27^3/6/9^2/6 x 9^4/6/9^4/6
which gave me: (27^3/6 x 9^4/6)/9
which I then simplified to (27³ x 9⁴)^1/6/9
Which I finally simplified to 4374/9=483

My problem is that 483 is too big. If i put 27^1/2/9^1/3 into the calculator, I get 2.498.
What did I do wrong?

27^1/2/9^{1/3 =} (3³)^1/2/(3²)^{1/3 =} 2.4980495329668129588263590875265

now use (3^a)^b =3^(ab)

 

[DrPhil]

I need help the the problem:
27^1/2/9^1/3
First, I multiplied 27^1/2 by 27^3/3 and 9^1/3 by 9^2/2
Then, I multiplied the resulting numbers: 27^3/6/9^2/6 x 9^4/6/9^4/6
which gave me: (27^3/6 x 9^4/6)/9
which I then simplified to (27³ x 9⁴)^1/6/9
Which I finally simplified to 4374/9=483

My problem is that 483 is too big. If i put 27^1/2/9^1/3 into the calculator, I get 2.498.
What did I do wrong?

To predict the order-of-magnidude of the result, make approximations.
27^1/2 ~ 25^1/2 = 5
9^1/3 ~ 8^1/3 = 2
approximate result = 5/2
That confirms that your calculator number is (probably) correct - and your analytic is not.

27^(3/3) is NOT the same as 27/27 - you can't just multiply the numerator by it, so that was where you began to go wrong. Look at the solutions where 27 is replaced by 3^3, and 9 is replaced by 3^2.

 

[stapel]

I need help the the problem:
27^1/2/9^1/3

What is the "problem"? Are you supposed to be simplifying to get an exact form? Simplify using radical notation? Find a decimal approximation? Or something else?

One method might be to convert everything to powers of 3:

27 = 3^3
9 = 3^2

27^(1/2) = (3^3)^(1/2) = 3^(3/2)
9^(1/3) = (3^2)^(1/3) = 3^(2/3)

Then [27^(1/2)] / [9^(1/3)] is 3^(3/2 - 2/3). And so forth.