# Complex Radical or Complex Root?

1. Sep 25, 2014

### iamjon.smith

1.5r250 (*r denotes the radical symbol)

in a radical expression where the index of the root is 1.5 and the number is 250

Use Calculator, Show Steps

2. Relevant equations

3. The attempt at a solution
First of all, using a calculator the answer comes out to 39.685, but I can't really show the steps since the calculator has an XsqrtY button and simply shows the answer. I attempted to find the steps as below, I know this is wrong since I took the square after breaking down the 250....please help.

1.5 r 250 = 1.5 r 10 * 25
1.5 * 5 r 10
7.5 r 10
7.5 * 3.162

I can't figure out the steps to solve this without the calculator, and I can't come up with the right answer. How do you solve a radical expression where the index is not 2 (square) or 3(cube), but 1.5 instead?

2. Sep 25, 2014

### SteamKing

Staff Emeritus
Are you familiar with exponents? How would you write the square root of 105 without using a radical but instead using an expression of the form 105^n. What value of n is required to express the square root of 105?

3. Sep 25, 2014

### LCKurtz

To do it by hand, write $1.5 = \frac 3 2$, use $a^{\frac 3 2} = \sqrt{a^3}$ and simplify.

[Edit:] Should be $a^{\frac 2 3} =\sqrt[3]{a^2}$

Last edited: Sep 25, 2014
4. Sep 25, 2014

### iamjon.smith

250^1/(3/2) = 250^2/3 = 39.685

Thanks for the direction!

5. Sep 25, 2014

### iamjon.smith

250^1/(3/2) = 250^2/3

6. Sep 25, 2014

### Staff: Mentor

This is completely unrelated to what SteamKing asked.

7. Sep 25, 2014

### Ray Vickson

What you do depends on what you mean. Do mean (1) $x = \sqrt[1.5]{250}$, or (2) $x = 250^{1.5}$? If you mean (1), you want to solve for $x$ that gives you $x^{1.5} = 250$, and the solution is $x = 250^{1/1.5} = 250^{2/3},$ which you can compute using logarithms (or by pressing a button if you have a calculator with a $x^n$ button for non-integer values of $n$). If you mean (2), you can either use logarithms, or else use the fact that 1.5 = 3/2 to write either $x = \sqrt{250^3}$ or $x = \sqrt{250}^3$; both formulas will give the same answer. Again, an alternative would be to use an $x^n$ button on a sufficiently powerful scientific calculator.