Question

Simplify the given expression. Show all steps. Submit your handwritten work or use the equation editor to receive credit.

(z√y√x)^8


`(√(z) √(y) √(x ) )^8`

Answer 1

`z^(4)`
`y^(4)`
`x^(4)`

Explanation:

Another way to write the expression is

(
`z^(1/2)`
`y^(1/2)`
`x^(1/2)`) ^8

the power to power principle tells us that when we raise a power to a power, we multiply the exponents

`(1)/(2)` x
`(8)/(1)` =
`(8)/(2)` = 4

`z^(4)`
`y^(4)`
`x^(4)`

Answer 2

`~\hspace{7em}\textit{rational exponents} \\\\ a^{( n)/( m)} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-( n)/( m)} \implies \cfrac{1}{a^{( n)/( m)}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\[-0.35em] ~\dotfill\\\\ \left( √(z) ~~ √(y) ~~ √(x) \right)^8\implies \left(√(zyx) \right)^8\implies \left(\sqrt[2]{(zyx)^1} \right)^8 \\\\\\ \left( (zyx)^{(1)/(2)} \right)^8\implies (zyx)^{(1)/(2)\cdot 8}\implies (zyx)^4\implies z^4 y^4 x^4`