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Some times in the books both of the below mentioned concepts are used interchangeably. Is there any reason for that? When to use -(2)^2 = -4 and (-2)^2. Explain with any useful examples.

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If the book is treating them as equivalent it is a mistake. In general $(ab)^r=a^rb^r$ therefore $ab^{\,r}\neq (ab)^r$ unless $a=0$ or $1$. In the example you stated, $-2^2=-1\times 2^2=-4$ whereas $(-2)^2=(-1)^2\times(-2)^2=4$

Essentially, a minus sign is equivalent to a "$...\times -1$" therefore (with $a>0$):

$\bullet$ it can be put inside or outside the brackets if the power is odd: $$(-a)^{2k+1}=(-1)\times(-1)^{2k}\times a^{2k+1}=-1\times1^k\times a^{2k+1}=-a^{2k+1}$$ $\bullet$ it can be disregarded completely if it is inside the brackets and the power is even: $$(-a)^{2k}=(-1)^{2k}\times a^{2k}=1^k\times a^{2k}=a^{2k}$$

$\bullet$ it must remain unchanged if it is outside the brackets and the power is even: $$-(a)^{2k}=-1\times a^{2k}=-a^{2k}$$

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The two expressions are not equal. The first is $-4$, since the square of $2$ is $4$ and its negation is $-4$. In the second, you multiply $-2$ by itself and get $4$. In short, the two can't be used interchangeably.

Here are some examples:

  • Your second one: $(-2)^2$. Here we apply the exponent to the parenthesized expression, so it's equal to $(-2)(-2) = 4$. We did the negation first, since it was grouped in parentheses.
  • Your first: $-(2)^2$. We still do the exponentiation first, giving us $(2)(2)=4$, and then negate the result, giving us $-4$.
  • $(1 + 2)^2$: we evaluate the term in parentheses first, to get $3$ and we square it to get $9$
  • $-(2)^3$: as above, we cube first (since the stuff in parentheses is already evaluated) and then we cube the result (since we evaluate exponents after we're finished evaluating the parens) to get $8$, then we negate the result (which is always done after we're done with the exponents) to get $-8$.
  • If you're okay so far, I'll leave with a test: what is $(-2)^3$. Interesting, eh?

I hope this was what you wanted. If you meant to ask

  1. When can you pull the minus sign inside of a parenthesis?
  2. Why is (-2)(-2) not equal to -4?

If you meant either of these two questions you should get a response within a few minutes. Good luck and welcome to the site!

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