# Simplifying $3^2 + (-8\div2)$

Simplify $3^2 + (-8\div2)$.

I know its not possible to divide $-8$ into $2$, so is there a way to make the 8 positive before working the problem?

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You are being asked to divide $-8$ by $2$, not into $2$. – Gerry Myerson Mar 12 '13 at 23:17
I believe that "by" and "into" are synonyms in this case. – Tanner Swett Mar 12 '13 at 23:18
@Tanner, that would be very sad, if true. – Gerry Myerson Mar 12 '13 at 23:19
Given the recent frequency of topics with the acronym "STEM" in them, educators apparently have decided that the problem with math education is that it needed a new, trendy name. – Emily Mar 12 '13 at 23:27
A Google search suggests that the word "into" in this case is ambiguous and thus should be avoided. – Tanner Swett Mar 12 '13 at 23:28

Hint: $(-8)\div2=-(8\div2)$. In fact, you should do division first before considering the minus sign.

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Thank you...that's the way I orginally worked the problem but wasn't sure.. – Deriah Mar 12 '13 at 23:30

On the contrary, it is possible to divide $-8$ by $2$. The definition of division is that $x \div y = z$ if and only if $x = z \cdot y$ and $y \neq 0$. We know that $-8 = -4 \cdot 2$; therefore, $-8 \div 2 = -4$.

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From http://en.wikipedia.org/wiki/Order_of_operations, the order of operations are as:

1. exponents and roots
2. multiplication and division

So, if you come across a problem like this, you should follow the steps above in order, from 1 to 3.

Now to address your question, for $3^2 + (-8÷2)$, we do the exponents first:

$$3^2 + (-8\div2)=3\times3+(-8\div2),$$ Then we can take the minus sign out of $8$ and multiply $3\times3$, so

$$3\times3+(-8\div2)=9+((-1)\times 8\div2).$$

Now, we can operate $8 \div 2$,

$$9+((-1)\times 8\div2)=9+((-1)\times 4).$$

Next, any number $\alpha$ multiplied by $(-1)$ will become $-\alpha$. So, if we have $\alpha$, then $\alpha\times(-1)=-\alpha$. For example if we have $\alpha=5$, then $5\times(-1)=-5$

So, we have $$9+((-1)\times 4)=9+(-4),$$

since here $\alpha=4$, so $4\times(-1)=(-1)\times 4=-4$.

Now, $$9+(-4)=9-4=5,$$

where we subtracted.

Hope this helps.

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