Why is sqrt(2)^3 = 2*sqrt(2)?

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tradBohus

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I couldn't seem to find other posts that ask this:
Why is sqrt(2)^3 = 2*sqrt(2)?

All I can imagine is that sqrt(2)^3=sqrt(8)=sqrt(4*2)=sqrt(4)*sqrt(2)=2*sqrt(2).
Is this true? If so, what rule says that sqrt(2)^3=sqrt(8)?

Thanks very much for your help!
 
You are correct. Basically it doesn't matter which you do first, the square root or the third power. Both are an exponent (the square root can also be written as to the power of 1/2), and just like multiplication exponents are commutative.

So if you take 2^3 (which equals 8) before doing the square root you get √8.
 
tradBohus said:
I couldn't seem to find other posts that ask this:
Why is sqrt(2)^3 = 2*sqrt(2)?

All I can imagine is that sqrt(2)^3=sqrt(8)=sqrt(4*2)=sqrt(4)*sqrt(2)=2*sqrt(2).
Is this true? If so, what rule says that sqrt(2)^3=sqrt(8)?

Thanks very much for your help!
Click to expand...
There are many ways to think of it. A simple one is \(\sqrt{2}^3=\sqrt{2}\sqrt{2}\sqrt{2}=(\sqrt{2}\sqrt{2})\sqrt{2}=2\sqrt{2}\)

But in general, \(\sqrt[m]{a}^n=\left(a^\frac{1}{m}\right)^n=a^{\frac{1}{m}\cdot n}=a^{n\cdot \frac{1}{m}}=\left(a^n\right)^\frac{1}{m}=\sqrt[m]{a^n}\).

So \(\sqrt{a}^3=\sqrt{a^3}\).
 
What is a definition of sqrt(2)? It is the number that when you multiply it by itself gives you 2. This the definition, nothing to understand!

As Dr Peterson pointed out [sqrt(2)]3 = sqrt(2) * sqrt(2) * sqrt(2) = (sqrt(2) * sqrt(2)) * sqrt(2) = 2sqrt(2)
 
[MATH](\sqrt{2})^3 = \sqrt{2} * \sqrt{2} * \sqrt{2} =[/MATH]
[MATH](\sqrt{2} * \sqrt{2}) * \sqrt{2} = (\sqrt{2})^2 * \sqrt{2} = 2\sqrt{2}?[/MATH]
 
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