Finding simplest radical form of a 4th root?

In summary, to find the simplest radical form of a 4√(x14), you need to move each occurrence of x^4 as a factor in the term of the radicand to outside the radical function. This results in a simplified form of x^3√x^2. However, if x is a negative real number or a complex number, simplifying further may not be well-defined and caution should be taken.
  • #1
Apollinaria
82
0
I haven't taken math in years and am having trouble understanding how to find simplest radical form of a 4√(x14).

I said x4√x10.

I realize I have 3 x4ths and x2 but I'm not sure if I can pull out more xs.

What are the rules for this? Ideas, insight?
 
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  • #2
Apollinaria said:
I haven't taken math in years and am having trouble understanding how to find simplest radical form of a 4√(x14).

I said x4√x10.

I realize I have 3 x4ths and x2 but I'm not sure if I can pull out more xs.

What are the rules for this? Ideas, insight?

For each occurance of x^4 as a factor in the term of the radicand, you have x to move to outside of the radical function.

[itex]\sqrt[4]{x^{14}}[/itex]=[itex]\sqrt[4]{x^4\cdot x^4\cdot x^4\cdot x^2}[/itex]

=[itex]x^3\sqrt[4]{x^2}[/itex]
 
  • #3
symbolipoint said:
For each occurance of x^4 as a factor in the term of the radicand, you have x to move to outside of the radical function.

[itex]\sqrt[4]{x^{14}}[/itex]=[itex]\sqrt[4]{x^4\cdot x^4\cdot x^4\cdot x^2}[/itex]

=[itex]x^3\sqrt[4]{x^2}[/itex]

Can you simplify [itex]\sqrt[4]{x^2}[/itex] still further to [itex]\sqrt{x}[/itex], or does that fall foul of something like principal roots?
 
  • #4
Hi Apollinaria!

There are a few rules for dealing with radical form, powers from powers, and sums of powers.
Here's how it works in your case:
$$\sqrt[4]{x^{14}} = (x^{14})^{\frac 1 4} = x^{14 \cdot \frac 1 4} = x^{3 + \frac 1 2} = x^3 \cdot x^{\frac 1 2} = x^3 \sqrt x$$
 
  • #5
sjb-2812 said:
Can you simplify [itex]\sqrt[4]{x^2}[/itex] still further to [itex]\sqrt{x}[/itex], or does that fall foul of something like principal roots?

That doesn't work if x is negative. If you are considering all the complex roots, then [itex]\sqrt[4]{x^2}[/itex] has 4 roots and [itex]\sqrt{x}[/itex] has 2.

But, I think it works if you group together the solutions like [itex]\sqrt[4]{x^2}[/itex] = [itex]\sqrt{x}[/itex] or [itex]\sqrt{-x}[/itex].
 
  • #6
sjb-2812 said:
Can you simplify [itex]\sqrt[4]{x^2}[/itex] still further to [itex]\sqrt{x}[/itex], or does that fall foul of something like principal roots?

Khashishi said:
That doesn't work if x is negative. If you are considering all the complex roots, then [itex]\sqrt[4]{x^2}[/itex] has 4 roots and [itex]\sqrt{x}[/itex] has 2.

But, I think it works if you group together the solutions like [itex]\sqrt[4]{x^2}[/itex] = [itex]\sqrt{x}[/itex] or [itex]\sqrt{-x}[/itex].

With the assumption that x is a non-negative real, it can be safely simplified.

If x can be a negative real, we have that √(x2) = |x| and 4√(x2)=√|x|.
However, in general we need to be very careful with negative real numbers and fractional powers.
They are generally not well-defined.
See for instance: http://en.wikipedia.org/wiki/Exponentiation#Rational_exponents
(The last couple of lines of the section.)

If x can be a complex number, it becomes even worse:
See for instance: http://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities
 

Related to Finding simplest radical form of a 4th root?

1. How do you find the simplest radical form of a 4th root?

To find the simplest radical form of a 4th root, you need to first determine the factors of the number under the radical sign. Then, group the factors in pairs and remove any perfect squares. The remaining factors will be the simplest radical form.

2. Can the simplest radical form of a 4th root be a decimal?

No, the simplest radical form of a 4th root should not be a decimal. It should only contain whole numbers and a square root symbol.

3. What if there are no perfect squares in the factors of a 4th root?

If there are no perfect squares in the factors of a 4th root, then the simplest radical form is the number under the radical sign with a 4th root symbol. For example, the simplest radical form of 12^(1/4) is 12^(1/4).

4. Is the simplest radical form of a 4th root always unique?

Yes, the simplest radical form of a 4th root is always unique. However, it may be written in different forms, such as with different factors or rearranged factors.

5. Can you simplify a 4th root that has a perfect 4th power under the radical sign?

No, if the number under the radical sign is a perfect 4th power, then it cannot be simplified any further. The simplest radical form would be the number itself.

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