regarding exponents, how to interpret and use.

Question

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.

Answer 1

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!

Answer 2

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: $$-(a)^{2k}=-1\times a^{2k}=-a^{2k}$$