Help: What is the answer to this math problem?

[Chrysalis Academy]

(-3^-3)^-2

 

I hope I wrote that equation correctly. I'm trying to write "parentheses negative 3 to the -3 power close parentheses to the -2 power"

 

So you have a -3 raised to the -3 power all inside of parentheses, raised to the -2 power.

 

The parentheses are really throwing me off. I think it should be (-3)^6 but the book says it is 3^6.  Help me understand why?

 

Or is it the same thing?

[creekland]

Any odd number to an even power is the same as the even number to that power.  Even powers cancel out the negatives through the multiplication.

 

(-3)^6 = 3^6

[Chrysalis Academy]

Ok, thank you. Would it be accurate to say that the expressions are equivalent, but 3^6 is the simplified form? 

 

Even though they are equivalent expressions, (-3)^6 wouldn't be accepted as the correct answer, right? Because it isn't simplified? 

 

Just trying to explain it clearly to my dd.  We get that they mean the same thing.

[creekland]

At our high school we would expect 3^6 as "the" correct answer for that problem if that were the problem by itself (testing knowledge of exponents), and yes, it's because it's simplified.

 

If it were part of any other problem, then the numerical answer of 729 would be expected. 

[Chrysalis Academy]

Thank you!

[lisabees]

Hmmm.  I would have gotten that wrong.

 

I would have said -3^6 or 729.

 

Thanks!

[MrSmith]

-3^6 is not equal to 729.

[creekland]

Hmmm.  I would have gotten that wrong.

 

I would have said -3^6 or 729.

 

Thanks!

 

Mr. Smith is correct.  -3^6 = -729 because of our order of operations.  Exponents come before negative signs, so what this means is take 3^6 = 729 and then make that negative.

 

(-3)^6 = 729 because parentheses come before exponents showing that I want -3 raised to the power.

 

Technically, the original problem, (-3^-3) comes first giving one (-1/27)^-2.

 

The next step gives me 729 because I square 27 (changing the sign) and flip the fraction due to the negative sign in the exponent.

 

Since multiplication & division are commutative, I can swap those orders and simply do (-3)^6.

 

Putting a question in like -3^6 is a common thing we do in tests to be certain students are up on their order of operations.  Of course, we teach it first.  We don't just surprise them on a test.

[lisabees]

I cannot thank you enough for the explanation.

 

Makes perfect sense.  I dropped the parentheses in my work.  Kind of important in math, eh?  

 

Rose - did you see this problem in Crocodiles and Coconuts?  We have seen similar in AoPS Pre-algebra.  Obviously, I rely on the solutions manual to clear things up for me.

 

 

[Chrysalis Academy]

The problem was on the Chapter 1 test in Chuckles the Rocket Dog (the third book).  He does a whole chapter on exponents right before jumping into Polynomials and Quadratics, which is what the rest of this book covers.  I'm so glad, because I had a feeling that this was something dd was kind of shaky on, particularly fractional and negative exponents.  She didn't do badly on the test, but we agreed that she's going to spend a few more days working with exponents before moving on in the book.  The parentheses thing, with negative exponents, was pretty confusing for her and while he does explain it, it's still kind of hard to get your head around.  I also really appreciate the detailed explanation that Creekland offered!

[MarkT]

Mr. Smith is correct.  -3^6 = -729 because of our order of operations.  Exponents come before negative signs, so what this means is take 3^6 = 729 and then make that negative.

 

(-3)^6 = 729 because parentheses come before exponents showing that I want -3 raised to the power.

 

Technically, the original problem, (-3^-3) comes first giving one (-1/27)^-2.

 

The next step gives me 729 because I square 27 (changing the sign) and flip the fraction due to the negative sign in the exponent.

 

Since multiplication & division are commutative, I can swap those orders and simply do (-3)^6.

 

Putting a question in like -3^6 is a common thing we do in tests to be certain students are up on their order of operations.  Of course, we teach it first.  We don't just surprise them on a test.

or apply the Power Rule of Exponents with parentheses around the base  (which is -3 in this case)

 

         (a^m)^n =  (a)^mn = (-3)^( -3 X -2) =  (-3)^6 

 

many text books show

 

     (a^m)^n =   a^mn  

 

which does not account for the sign of a so this could be confusing to students

[lisabees]

The problem was on the Chapter 1 test in Chuckles the Rocket Dog (the third book).  He does a whole chapter on exponents right before jumping into Polynomials and Quadratics, which is what the rest of this book covers.  I'm so glad, because I had a feeling that this was something dd was kind of shaky on, particularly fractional and negative exponents.  She didn't do badly on the test, but we agreed that she's going to spend a few more days working with exponents before moving on in the book.  The parentheses thing, with negative exponents, was pretty confusing for her and while he does explain it, it's still kind of hard to get your head around.  I also really appreciate the detailed explanation that Creekland offered!

 

I just spent the last hour reviewing again with dd.

 

Clearly, one must be very careful with these problems. 

 

I had dd talk through each problem, explaining how she got to the answer.

 

Wouldn't you know that she got every single problem correct doing it this way?!

[Chrysalis Academy]

or apply the Power Rule of Exponents with parentheses around the base  (which is -3 in this case)

 

         (a^m)^n =  (a)^mn = (-3)^( -3 X -2) =  (-3)^6 

 

many text books show

 

     (a^m)^n =   a^mn  

 

which does not account for the sign of a so this could be confusing to students

 

Yes, I think she gets the power rule in general but is kind of confused with the combo of the parentheses and the negative sign.  She did just what you showed in your first step, but stopped there, rather than taking the next step to simplify (-3)^6 into 3^6.  As soon as we talked about it, she was very clear that they were the same thing - she understand that raising a negative number to a positive (ETA: EVEN) power has a positive solution. But she was expressing confusion at the distinction between

 

(-3)^6 and -3^6, for example.  My explanation was  that -3^6 is like -1*3^6, so that by order of operations you do the exponent computation first, and then multiply it by -1. Right?

 

I think she just needs more practice with these so that she can really see the parentheses and negative signs for what they are. So it is probably more of an issue with applying Order of Operations to exponent problems, rather than a misunderstanding of the exponent rules.

[creekland]

Yes, I think she gets the power rule in general but is kind of confused with the combo of the parentheses and the negative sign.  She did just what you showed in your first step, but stopped there, rather than taking the next step to simplify (-3)^6 into 3^6.  As soon as we talked about it, she was very clear that they were the same thing - she understand that raising a negative number to a positive power has a positive solution. But she was expressing confusion at the distinction between

 

(-3)^6 and -3^6, for example.  My explanation was  that -3^6 is like -1*3^6, so that by order of operations you do the exponent computation first, and then multiply it by -1. Right?

 

I think she just needs more practice with these so that she can really see the parentheses and negative signs for what they are. So it is probably more of an issue with applying Order of Operations to exponent problems, rather than a misunderstanding of the exponent rules.

 

A positive EVEN power.  A negative raised to a positive ODD power still leaves a negative answer.

 

(-3)^5 = -243

 

Powers are all multiplication - in this case - (-3) multiplied by itself 5 times.  Even the power rule is derived from the basic multiplication.

 

And yes, without parentheses the multiplication by -1 after the exponent is correct due to our order of operations.

[Chrysalis Academy]

Right, thank you!  It is a good lesson about how precise you need to be, and how focused on the details, for math! Not always my strong suit.

[MrSmith]

Yes, I think she gets the power rule in general but is kind of confused with the combo of the parentheses and the negative sign.  She did just what you showed in your first step, but stopped there, rather than taking the next step to simplify (-3)^6 into 3^6.  As soon as we talked about it, she was very clear that they were the same thing - she understand that raising a negative number to a positive (ETA: EVEN) power has a positive solution. But she was expressing confusion at the distinction between

 

(-3)^6 and -3^6, for example.  My explanation was  that -3^6 is like -1*3^6, so that by order of operations you do the exponent computation first, and then multiply it by -1. Right?

 

I think she just needs more practice with these so that she can really see the parentheses and negative signs for what they are. So it is probably more of an issue with applying Order of Operations to exponent problems, rather than a misunderstanding of the exponent rules.

 

Sometimes when students are knee deep in the details it is hard to tell if they understand the concepts, even when they properly apply the material.  When we were doing AOPS Pre-A chapter 1, my kid was consistently answering correctly the end of section exercises.  It wasn't until the chapter review (and especially the Challenge section) that we saw he wasn't really getting it.

 

So... in keeping with the spirit of 'let's torture Boy whenever possible' , I copied all the properties onto a sheet of paper (there are a lot...).  I made him re-work each problem, citing a property for each step he performed.  He hated math for about 3 weeks :D, but by the end we both knew all his mistakes were silly errors and not conceptual errors.

 

At the same time, he learned that if he does not understand a concept, it is best to ask Dad ASAP rather than waiting for me to notice.

[EndOfOrdinary]

There are AoPS videos on this very subject! They were funny and helpful, classic Rick. My son found them to ve especially helpful as a visual learner to watch someone else go through the process.

[Chrysalis Academy]

There are AoPS videos on this very subject! They were funny and helpful, classic Rick. My son found them to ve especially helpful as a visual learner to watch someone else go through the process.

 

Shannon watched the videos yesterday, and finally turned this section blue on Alcumus! We went over a bunch of problems together too. I think she basically has it down but needs to practice, practice, practice. The alcumus problems cleverly test your ability to apply all of the exponent rules. I was showing her that if she was calculating a big number, she was missing the point - she needed to look more closely at the problem and see where she could apply the rules to simplify it.

 

Honestly, I just think she needs more practice to solidify the rules and make applying them automatic. Does anyone have suggestions for where I could go for this kind of practice problems? Exponent problems that test order of operations and include both negative bases, negative exponents, fractional exponents, and exponents in fractions.

[MarkT]

 

Honestly, I just think she needs more practice to solidify the rules and make applying them automatic. Does anyone have suggestions for where I could go for this kind of practice problems? Exponent problems that test order of operations and include both negative bases, negative exponents, fractional exponents, and exponents in fractions.

https://www.kutasoftware.com/free.html

 

Algebra 1

 

select Properties of exponents  -  Hard

[Chrysalis Academy]

https://www.kutasoftware.com/free.html

 

Algebra 1

 

select Properties of exponents  -  Hard

 

Thank you!  This is exactly what I was looking for. And I'm happy to have this as a resource for future practice problems if we need them. I really like the Arbor math books for Shannon, it's a teaching style she clicks with so well, and I like Alcumus for challenge problems. But sometimes, like Mr Smith said, you just need to hammer on something until they really get it down, and I'm often at a loss for sources of problems in that situation.

[lisabees]

Oooh, that was fun, MarkT!

 

Tomorrow, I will let dd have fun with it!

[Chrysalis Academy]

Yeah, those worksheets were super helpful in isolating exactly what her trouble was - it was parentheses and properly distributing the exponent when the base was in parentheses.  A few issues with dealing with negative exponents in the numerator or denominator of a fraction, too.  Great for identifying and addressing the shaky concepts!  Thanks again.

[lisabees]

And, Rose, don't forget to pull out AoPS Pre-algebra and use the exercises for extra practice.  We use MEP Secondary 7-9 every day for that kind of practice.

 

We also go through dd's incorrect answers on alcumus, too, using a whiteboard.  

 

Why does that always seem more helpful than paper?  Is it because it is just more fun?! :lol:

[Chrysalis Academy]

And, Rose, don't forget to pull out AoPS Pre-algebra and use the exercises for extra practice.  We use MEP Secondary 7-9 every day for that kind of practice.

 

We also go through dd's incorrect answers on alcumus, too, using a whiteboard.  

 

Why does that always seem more helpful than paper?  Is it because it is just more fun?! :lol:

 

I was looking at both the PreA and Algebra books yesterday! We may work through more of those sections if we need to.