What is wrong?

Here is a "proof" that $-1 = 1$ :

• 1. $1 = \sqrt { { 1 }^{ 2 } }$

We know that:

• 2. ${ 1 }^{ 2 } = { \left( -1 \right) }^{ 2 }$

Substituting the 1. through the 2. leads to:

• 3. $\sqrt { { 1 }^{ 2 } } =\sqrt { { \left( -1 \right) }^{ 2 } }$

Due to the power law $\sqrt [ m ]{ { a }^{ n } } = { a }^{ \frac { n }{ m } }$ we get:

• 4. $\sqrt { { \left( -1 \right) }^{ 2 } } ={ \left( -1 \right) }^{ \frac { 2 }{ 2 } }$

And finally simplifying the result.

• 5. ${ \left( -1 \right) }^{ \frac { 2 }{ 2 } }{ ={ \left( -1 \right) }^{ 1 }=-1 }$

  Thus $-1 = 1 \quad q.e.d$

Which step is the first one, which is wrong?