Understanding Cube Roots.

[jonboy]

Hello! I do not understand why for: $\displaystyle \,\sqrt[3]{8} = \sqrt[3]{{2 \bullet 2 \bullet 2}} = 8\,$ taking $\displaystyle \,\sqrt{8}^3\approx 22.6\,$ equals a different quantity. Isnt $\displaystyle \,\sqrt[3]{8}$ the same thing as $\displaystyle \sqrt{8}^3$? If not can someone explain the difference for me (my tutor is not available till' 2). Any help is greatly appreciated.

 

[stapel]

Is taking the square root the same as squaring? So why would taking the cube root be the same as cubing?

Meanwhile, the cube root of eight is not eight; it is two. And the square root of eight cubed is not the same as the cube root of eight.

Eliz.

 

[jonboy]

So what does the superscript 3 in:$\displaystyle \;\sqrt[3]{8}\;$ mean if it does not mean $\displaystyle \sqrt{8}\bullet\sqrt{8}\bullet\sqrt{8}$?

 

[galactus]

It means \(\displaystyle \L\\8^{\frac{1}{3}}\), just the same as \(\displaystyle \L\\\sqrt{8}\) means \(\displaystyle \L\\8^{\frac{1}{2}}\). We just, normally, omit the superscript 2 in the radical sign.

\(\displaystyle \L\\\sqrt[4]{8}=8^{\frac{1}{4}}\)

\(\displaystyle \L\\\sqrt[5]{8}=8^{\frac{1}{5}}\)

etc, etc, etc.....

In your last post, you are cubing the square root of 8. That's also written

as\(\displaystyle \L\\8^{\frac{3}{2}}\). See?. Not the same thing as \(\displaystyle \L\\8^{\frac

{1}{3}}\)

 

[jonboy]

Ok here is how I view it: $\displaystyle \;\sqrt[3]{2^3}=2\;$ Since the cube root and sqrt 3 cancle each other out.

Thank you Stapel and Galactus for you help.

 

[tkhunny]

sqrt 3

I think you are confused only by the LaTeX notation. It is unfortunate that a CUBE ROOT is coded as sqrt[3]. Don't be confused by that. A cube root is not just a different dialect from a square root. It is a rather different animal.

 

[jonboy]

sqrt 3

I think you are confused only by the LaTeX notation. It is unfortunate that a CUBE ROOT is coded as sqrt[3]. Don't be confused by that. A cube root is not just a different dialect from a square root. It is a rather different animal.

Yeah. Thank you for correcting me!