Question

How would you simplify `root(4)(32x^4z^19)`?

Answer 1

`2x\root[4]{2z^{19}}`

Explanation:

`\root[4]{32x^4z^{19}}`

Concepts:
`\root[a]{x^b}=x^{b/a}` or `x^{b\diva}`

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Work: Let's split this up.

Part one: `\root[4]{32}=\root[4]{8\cdot4}=\root[4]{(4xx2)\cdot(2xx2)}=\root[4]{((2xx2)xx2)\cdot(2xx2)}`
take out four 2's and you have one 2 remaining: `2\root[4]{2}`
Part two: `\root[4]{x^4}=x^{4\div4}=x^1`, or `x`
Part three: `\root[4]{z^{19}}=z^{19\div4}` which does not get you a whole number exponent, so let's keep it under the `4^{th}` root. `\root[4]{z^{19}}`

Putting it all together: `2\root[4]{2}\cdotx\cdot\root[4]{z^{19}}=2x\root[4]{2z^{19}}`

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Source:
Symbolab Step-by-Step