I'm looking for a really intuitive explanation of why the radical rules work. I've gone through this process with exponent rules (like those below) and they make complete intuitive sense to me after working them out. I've also been able to find a lot of resources online about exponent rules (many provided by people on this forum - thank you), but not much at all about radicals/roots.
These make sense:
[math](\frac{a}{b})^2 = (\frac{a}{b})*(\frac{a}{b}) = \frac{a*a}{b*b} = \frac{a^2}{b^2}[/math]
or
[math](ab)^2 = (ab)*(ab) = (a*a*b*b) = (a*a)(b*b) = a^2b^2[/math]
But can't find much on ones such as:
[math]\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}[/math]
or
[math]\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]a}{\sqrt[n]b}[/math]
My book gives the rules, but not explanations for WHY these things are true. Any explanations or even links to good outside sources would be tremendously appreciated.
These make sense:
[math](\frac{a}{b})^2 = (\frac{a}{b})*(\frac{a}{b}) = \frac{a*a}{b*b} = \frac{a^2}{b^2}[/math]
or
[math](ab)^2 = (ab)*(ab) = (a*a*b*b) = (a*a)(b*b) = a^2b^2[/math]
But can't find much on ones such as:
[math]\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}[/math]
or
[math]\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]a}{\sqrt[n]b}[/math]
My book gives the rules, but not explanations for WHY these things are true. Any explanations or even links to good outside sources would be tremendously appreciated.
