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The value of $d^2 =(-4)^2= (-4) \cdot (-4) =16$, so how come $a^3= (-2)^3= (-2) \cdot (-2) \cdot (-2)= -8$?

Why does the first one remain a positive number and the second one a negative? Please explain clearly I'm a bit thick. What are the rules?

Thomas
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greg
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5 Answers5

3

Multiplying two positive numbers results another positive one: $1*1=1$, I think it's clear.

Multiplying one positive with one negative results a negative one.

  • You take one times -1, which is -1: $1*-1=-1$.
  • Or you take the -1 times (take the opposite of) 1, which is also -1: $-1*1=-1$

Multiplying two negatives results in positive: you take the opposite of a negative number, which is positive: $-1*-1=1$

Multiple multiplication isn't defined in mathematics, you always multiply two numbers, then multiply the previous result with the next number, so:

$a^3=(-2)^3=-2*-2*-2$

At first multiply only two numbers:

$=(-2*-2)*-2=+4*-2$

Then multiply the rest:

$=(+4)*(-2)=-8$

Máté Juhász
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1

Basic rules (where * means multiplication)

$+ * + = +$

$- * - = +$

$+ * - = -$

$- * + = -$ You used this rule in $-4*-4=+16$

Now,

$$ -2 * -2 * -2$$ is:

$$ (-2 * -2) * -2$$ is:

$$ (+4) * -2 \; Here\; we\; used\; the\; rule: \; - * - = +$$

Now apply the rule: $+ * - = -$ to get the result -8

NoChance
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1

The rules are

  • positive times positive is positive
  • positive times negative is negative
  • negative times positive is negative
  • negative times negative is positive

Why are these rules true? This has been asked before ans you can find the answer here:

Why is negative times negative = positive?

So $(-4)\cdot(-4) = 16$ because you have the product of two negative numbers. And $4\cdot (-4) = -16$ because you have the product of a positive number with a negative number.

$(-2)^3 = -8$ is negative because you have the product of three negative numbers. Think about it this way: negative times negative times negative is (negative times negative) times negative is positive times negative is negative. Or $$ -\cdot -\cdot - = (-\cdot -) \cdot - = + \cdot - = -. $$

Community
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Thomas
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  • Ok so, negative times negative is positive so does that mean negative times negative times negative is positive because theyre are 3 negatives? – greg Oct 08 '15 at 12:50
  • @greg: no, neg. times neg. times neg. is (neg. times neg.) times neg. is pos. times neg. is neg. Does that make sense? – Thomas Oct 08 '15 at 12:55
  • Thanks i think so. So what exactly is it that makes the answers so different? I thought it was the fact that there are three negatives which made it negative. – greg Oct 08 '15 at 13:03
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    Ok i finally got it. Thanks alot :) – greg Oct 08 '15 at 13:07
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Remember this thumb rule: + times + is positive - times+ is negative - times - is positive Now try solving by using two numbers at once. (-2)³=(-2).(-2).(-2) . Take the first two numbers: - and - make a positive sign. Now that positive, when multiplied with a '-' gives negative. Thus, -8 is obtained.

Kishore Pattabhiraman
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That's a consequence of that $a\times (-b)$ and $(-a)\times b$ is defined as $-(a\times b)$. This means that $(-a)^2 = (-a)\times(-a) = a\times a = a^2$, but $(-a)^3 = (-a)\times(-a)\times(-a) = -(a\times a\times a) = -a^3$.

But to dig deeper in why one would define it so, we have to consider the properties of multiplication when the factors are positive (ie the algebraic laws). Consider especially the distributive laws that say $(a+b)c = ac+bc$, what if we would allow factors to be negative? Then we would need to have that $(a + (-a))c) = ac + (-a)c$, but as $a+(-a)=0$ it would be required that $ac + (-a)c = 0$, that is $(-a)c = -(ac)$.

It just happens that this definition is just what's needed to get the laws to remain if we allow factors to be negative.

skyking
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