Simplifying Radicals

[GWS]

Can someone explain how to simplify tricky radicals like the following:

1) Simplify the cube root of {(x-5)^5}

2) Simplify the sqrt{(x-6)^6}

3) Simplify the cube root of {(64^9)(y^10)(z^4)}

 

[stapel]

These aren't any trickier than, say, "cbrt[x⁵] = x cbrt[x²]". Follow exactly the same methods and rules.

Eliz.

 

[pka]

Note in the first one there are no domain issues:
\(\displaystyle \L
\sqrt[3]{{\left( {x - 5} \right)^5 }} = \left( {x - 5} \right)\sqrt[3]{{\left( {x - 5} \right)^2 }}\)

But in the second we have to be mindful of domain:
\(\displaystyle \L
\sqrt {\left( {x - 6} \right)^6 } = \left| {x - 6} \right|^3\)

Again no domain issues:
\(\displaystyle \L
\sqrt[3]{{\left( {64} \right)^9 y^{10} z^4 }} = \left( {64} \right)^3 \left( {y^3 z} \right)\sqrt[3]{{yz}}\)