# What is the difference between √-4^2 and √(-4)^2

### What is the difference between √-4^2 and √(-4)^2

So $$\sqrt{-4^2}$$ simplifies to an imaginary number and $$\sqrt{(-4)^2}$$ simplifies to a real number.

How and why does this happen?
Guest

### Re: What is the difference between √-4^2 and √(-4)^2

$$-1 = i^2$$
$$\sqrt{-4^2} = \sqrt{-1 \cdot 4^2} = \sqrt{i^4 \cdot 2^2} = \sqrt{(4i)^2} = 4i$$
Guest

### Re: What is the difference between √-4^2 and √(-4)^2

$$\sqrt{(-4)^2} = \sqrt{(4)^2} = 4$$

Is it clear?
Guest

### Re: What is the difference between √-4^2 and √(-4)^2

(-4)^2= (-4)(-4)= 16

-4^2= -(4)(4)= -16.

It is a question of "precedence of operators". Without the parentheses, you first square 4, getting 16 then take the negative of 16 to get -16.

With the paretheses you do what is in the parentheses, to get -4, then square that. The square of -4 is 16.
Guest

### Re: What is the difference between √-4^2 and √(-4)^2

Guest wrote:(-4)^2= (-4)(-4)= 16

-4^2= -(4)(4)= -16.

It is a question of "precedence of operators". Without the parentheses, you first square 4, getting 16 then take the negative of 16 to get -16.

With the paretheses you do what is in the parentheses, to get -4, then square that. The square of -4 is 16.

This makes sense. Thanks!
Guest