Exponents and negative numbers

Why do I get a negative answer even though the exponent is even and the other exponent is odd? Doesn't the exponent being odd or even determine if the answer is negative or positive?

Here are two examples:

−(3/2)² = −2 1/4.
−(3/2)³ = −3 3/8.

This is really a matter of notation.

We have agreed that −(3/2)² is not the same as (-3/2)².

If you want to calculate -3/2 to the second power, it is written so that -3/2 is inside parenthesis, like this: (-3/2)².

The answer to this is indeed positive: it is (-3/2) × (-3/2) = 9/4 = 2 1/4.

However, if the minus sign is outside the parenthesis, it means the opposite number of whatever it's in front of.

So, −(3/2)² means: "First calculate (3/2)², then take the opposite of that."
First you calculate (3/2)² = 9/4, and its opposite is -9/4.

In other words, the location of the parenthesis makes a huge difference.

With exponent 3, we get:

−(3/2)³ = −(3/2 × 3/2 × 3/2) = −27/8.

(-3/2)³ = (-3/2) × (-3/2) × (-3/2) = -27/8 or -3 3/8.

So in this case, these two expressions have the same value.

An advanced student may notice:

If n is even, and x is any real number, then

(-x)ⁿ = xⁿ ; in other words it won't matter if you calculate the power using x or using negative x - the result is the same.

And if n is odd, then

(-x)ⁿ = −xⁿ ; in other words, it's like you can "pull the minus out" from inside those parenthesis and put it out front.

The World Math Day

In the World Math Day, children from across the globe unite in their quest to set a world record in answering mental arithmetic questions. This event involves more than a million students from 160 countries.

Why should your or my children participate? Well, they might love being part of setting a world record! And it's just simple math questions!

Also good to know:

* The event is designed for all ages and ability levels.
* There are prizes.
* And it's FREE!

How will it work?
Simply register your kids online. The actual event will take place in real-time. Kids will answer simple arithmetic questions and play live with other students around the globe.

In 2008 more than one million children from 160 countries combined to correctly answer 182,455,169 questions. This year, they try to break this record.

World Math Day takes place on 4 March, 2009. Please register here: www.WorldMathDay.com.

The Human Calculator Scott Flansburg, the Guiness Record Holder for "Fastest Human Calculator", has been appointed the Global Patron for World Math Day 2009.

Update: The world record was broken. The 1,952,879 students who took part correctly answered 452,681,681 arithmetic questions. Kaya Genc, an 11-year old student from Australia, ended up being the World Math Day Champion for 2009 by answering correctly 129,106 questions in 48 hours.






The LEVELS

I received some more information about World Math Day levels and ages.

Level 1    Ages 5-7 Addition within 0-10
             
Level 2    Ages 8-9 Addition within 0-20, Subtraction within 0-10

Level 3    Ages 10-11 Addition within 0-99, Subtraction within 0-20
                      Multiplication tables of 2 and 5

Level 4    Ages 12+ Addition and subtraction within 0-99
 basic multiplication and division facts

If you want your child to be on level 4, sign her up with an age 12 or older. Once you sign up, these levels cannot be changed. So if your child is ahead in grade levels, put in the child's "age" based on the levels explained above.

I registered my oldest daughter. She's enjoyed competing against other kids around the globe. Each competition takes 1 minute and the program finds 2-3 other kids to compete against. She also likes that you can choose the hair color, style, and hat or other head piece for your virtual character.

The Date
As the event runs for as long as it is 4 March somewhere in the world, this means the event runs for a total of 48 hours. World Math Day commences when it is midnight in New Zealand – being 6.00am 3 March in Washington.

Multiplication tables video

This video is about using a structured drill to learn multiplication tables. I consider "structured" drill far superior to starting kids with random drill. This kind of drill uses the structure of the table itself to help the children at first.

Once they master this, it's time to use random drills (flashcards).