Question

Why is (-8)^1/2 not equal to -(8)^1/2
but, (-27)^1/3 = -(27)^1/3 ?

Answer 1

\[\left( -8 \right)^{\frac{ 1 }{ 2 }}=\left( 4*2*-1 \right)^{\frac{ 1 }{ 2 }}=2\sqrt{2}\iota \]
\[-\left( 8 \right)^{\frac{ 1 }{ 2 }}=-\left( 2^2*2 \right)^{\frac{ 1 }{ 2 }}=-2\sqrt{2}\]
\[\left( -27 \right)^{\frac{ 1 }{ 3 }}=\left\{ \left( -3 \right)^3 \right\}^{\frac{ 1 }{ 3 }}=-3\]
\[-\left( 27 \right)^{\frac{ 1 }{ 3 }}=-\left\{ \left( 3 \right)^3 \right\}^{\frac{ 1 }{ 3 }}=-3\]

Answer 2

Thank you. ^-^

Answer 3

yw