Question

Provide a clear answer while explaining how to do each step to the question below:

Which of the following expressions are equivalent to
`\sqrt[3]{8x^5y^7}`?

A.
`8{^3}x^(15) y^(21)`
B.
`8^(3)x^(5)/(3)y^(21)/(3)`
C.
`2x^(15)y^(21)`
D.
`2x^(5)/(3)y^(7)/(3)`

Answer 1

`\bf ~\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] \rule{34em}{0.25pt}\\\\ \sqrt[3]{8x^5y^7} \\\\[-0.35em] ~\dotfill\\\\ \sqrt[3]{8x^5y^7}\implies \sqrt[3]{2^3x^5y^7}\implies \left( 2^3x^5y^7 \right)^{(1)/(3)}\implies 2^{3\cdot (1)/(3)}x^{5\cdot (1)/(3)}y^{7\cdot (1)/(3)}\implies 2x^{(5)/(3)}y^{(7)/(3)}`