Question

Which expression is equivalent to (4^5/4*4^1/4 divided by 4^1/2) ^1/2

Answer 1

`\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^(-n) \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^(-n)} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^(-m)\implies a^(n-m) \\\\\\ ~\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}`

`\bf \left( \cfrac{4^{(5)/(4)}\cdot 4^{(1)/(4)}}{4^{(1)/(2)}} \right)^{(1)/(2)}\implies \left( \cfrac{4^{(5)/(4)\cdot (1)/(2)}\cdot 4^{(1)/(4)\cdot (1)/(2)}}{4^{(1)/(2)\cdot (1)/(2)}} \right)\implies \cfrac{4^{(5)/(8)}\cdot 4^{(1)/(8)}}{4^{(1)/(4)}}\implies \cfrac{4^{(5)/(8)+(1)/(8)}}{4^{(1)/(4)}}`

`\bf \cfrac{4^{(6)/(8)}}{4^{(1)/(4)}}\implies \cfrac{4^{(3)/(4)}}{4^{(1)/(4)}}\implies 4^{(3)/(4)}\cdot 4^{-(1)/(4)}\implies 4^{(3)/(4)-(1)/(4)}\implies 4^{(2)/(4)}\implies 4^{(1)/(2)}\implies √(4)\implies 2`