The square root of one-fifth minus the square root of 5

If you mean the fifth root, use the xth-root function (MATH 5). For example, to calculate the fifth root of 32, press 5 MATH 5 3 2 ENTER

Square root equivalent to power 1/2
Fifth root equivalent to power 1/5
square root of fifth root (if argument of square root is positive) is equivalent to power 1/10
64^(1/10)*q^(4/5)*v^(3/5)
using 64=2^6 you get 2^(3/5)*q^(4/5)*v^(3/5)

You should be careful with the symbols you are using. Roots are different. When you say `5 square root symbol 1.734` you are in fact doing the following: 5x square root(1.734) which is 6.5850.
What you need to do is extract the fifth root of 1.734.
The key to use to extract a root of index n (in your case 5) is to press the digit key for the index, then press [2nd][Y^x]. On the calculator near the [y^x] key you can see xRoot. This is the universal root key.

(Square root(7))*(Square root (15))=(Square root (7*15)=Square root(105)

Use the xth-root function in the MATH menu (it should be the fifth item). For example, to calculate the 15th root of 2, press 1 5 MATH 5 2 ENTER .

Two ways:

2ND [square-root] ( 8 7 . 3 4 / 5 ) ENTER

8 7 . 3 4 / 5 ENTER then 2ND [square-root] 2ND ANS ENTER

The first sequence takes the square root of 87.34/5, the second sequence calculates 87.34/5 then takes the square root of the result.

Use the xth-root function to compute any root. It's the fifth entry on the MATH menu, which you can get by pressing the MATH key. The fourth entry on the same menu gives you the third root without needing the 3 to be specified.

For example, to compute the 5th root of 32, press 5 MATH 5 3 2 ENTER.

1. Use the universal power key marked with [^].
   
2. To avoid problems if the exponent is an expression, enclose it in parentheses.
3. To enter any power type number^(exponent)
4. If exponent is negative use the (-) change sign key next to the dot, below the 3 key.
5. To calculate the roots (cubic, fourth, fifth, etc.) roots use the fact that a root of n-th order can be represented as ^(1/n)
6. Ex: cube root of 27: 27^(1/3); square root of 64 : 64^(1/2) or 64^(0.5)
   

You can enter any root by typing the number, then hitting MATH and 5, which brings up the root symbol with the x in front of it: x√. So the fifth-root would be 5 -> MATH -> 5 and then whatever number you want to get the fifth-root for: 5x√10 for example. As someone else had mentioned, you can also raise it to a rational power: 3^(1/3) which would be the same as the cubed root of 3, but you could also type: 3 -> MATH -> 5 -> 3 and get the same answer, but looking like this in your calculator: 3x√3. The option for 4 actually is a predefined cubed-root, and the one for option 5 there is the root symbol that can be used with any number before it to get any root you want. There are no parenthesis as you get when using the predefined square-root and cubed-root functions, though, so you may want to type them in yourself if entering a long string of operations in the calculator at one time to make sure the calculator doesn't include numbers under the root that you don't want it to. But I guess it depends on preference in terms of what method you choose between the rational exponents or the root symbol (and most would go with whichever seems easiest and quickest to enter), but you asked specifically how to get the cubed-root on the TI-83 Plus, so there's my best attempt at answering your question.

There is a way to do it. I believe you go into "complex" or "math" buttons. These buttons are yellow on the calculator. Therefore, to access them you must hit "2nd" then the button.

Better yet, you can also get around this dilemma another way. You can enter "the cubed root of x" by raising x to 1/3.

For example the cubed root of x = x^(1/3). It is best to place parentheses around 1/3 so the calculator knows exactly what you mean.

Another example, the "cubed root of (x + 1)" can be entered by:
(x+1)^(1/3) Note the parenteses around both (x+1) and (1/3). This applies if the radical cover both "x" and "1".

Hope this helps.