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Click to Select a Page... Main Page Introduction Comparing The Math Operators I Comparing The Math Operators II Using Numbers 3 and Higher 1 ✱ b a ✱ 1 a ✱ 0 0 ✱ b a ✱ -b -a ✱ b -a ✱ -b Formula Summary So Far Integer a > 1 ✱ Decimal b > 1 Integer a > 1 ✱ Decimal 0 < b < 1 Integer a > 1 ✱ Decimal b < 0 Decimal a > 1 ✱ Decimal b > 1 Decimal a > 1 ✱ Decimal 0 < b < 1 Decimal a > 1 ✱ Decimal b < 0 Decimal 0 < a < 1 ✱ Decimal b > 1 Decimal 0 < a < 1 ✱ Decimal 0 < b < 1 Decimal 0 < a < 1 ✱ Decimal b < 0 Decimal -2 < a < 0 ✱ Decimal -1 < b < 2 Formula Summary So Far II Why 2 ✱ 0 is Not 1 Finding The Inverse Operations 1.25 ✱ -10 to 10 1.5 ✱ -10 to 10 1.75 ✱ -10 to 10 2 ✱ -10 to 10 2.5 ✱ -10 to 10 3 ✱ -10 to 10 3.5 ✱ -10 to 10 4 ✱ -10 to 10 4.5 ✱ -10 to 10 5 ✱ -10 to 10 6 ✱ -10 to 10 7 ✱ -10 to 10 8 ✱ -10 to 10 9 ✱ -10 to 10 10 ✱ -10 to 10 Fun With Math Strip Home Page DavesFunStuff Dave's Fun Stuff Find your favorite TV shows with "Let's Watch TV!" Dave's Fun Stuff SDN Media News and More Dave's Fun Stuff Super Birthdays Contact Webmaster

Why 2 ✱ 0 is Not 1

You may be wondering why a ✱ 0 is not equal to 1.

Let's go fill the "a" with the value of 2 and the "b" with the value of 3 like this:

2 ✱ 3

As you have learned, 2 ✱ 3 is taking 2 to the exponent of 2, then taking that answer to the power of 2. Like this:

(2 ^ 2) ^ 2 = 4 ^ 2 = 16.

If you go one integer down in the value of "b" to be the value of 2 like this:

2 ✱ 2

Then you take the number 2 to the exponent of 2 and you get 4.

2 ^ 2 = 4

The number 4 is the principal square root of 16.

2 ^ 2 = 4 = √16

And likewise if you're going forward with "b":

(2 ^ 2) ^ 2 = 16 = 4²

Now let's let "b" = 4.

((2 ^ 2) ^ 2) ^ 2 = (4 ^ 2) ^ 2 = 16 ^ 2 = 256.

And to summarize the relationships of the answers below:

2 ✱ 2 = 4 = 4¹ = ¹√4 = ²√16 = 16^(1/2) = ⁴√256 = 256^(1/4)

2 ✱ 3 = 16 = 4² = ^(1/2)√4 = ¹√16 = 16¹ = ²√256 = 256^(1/2)

2 ✱ 4 = 256 = 4⁴ = ^(1/4)√4 = ^(1/2)√16 = 16² = ¹√256 = 256¹

As you can see, each of the answers has a relation with the answers above and below as the value of "b" changes. The values of the exponents and degrees change in relation to the other answers in a predictable pattern. 4 is the square root of 16 and the 4th root of 256. 16 is 4 squared and the square root of 256. 256 is 4 to the exponent of 4 and 16 squared.

Now let's take it further down and let "b" be 1 to 3.

2 ✱ 1 = 2 = 2¹ = ¹√2 = ²√4 = 4^(1/2) = ⁴√16 = 16^(1/4)

2 ✱ 2 = 4 = 2² = ^(1/2)√2 = ¹√4 = 4¹ = ²√16 = 16^(1/2)

2 ✱ 3 = 16 = 2⁴ = ^(1/4)√2 = ^(1/2)√4 = 4² = ¹√16 = 16¹

The pattern contunes. 2 is the square root of 4 and the 4th root of 16. 4 is 2 squared and the square root of 16. 16 is 2 to the exponent of 4 and 4 squared.

Now, let's take it further down and let "b" be from 0 to 2.

2 ✱ 0 = ²√2 = (²√2)¹ = ¹√(²√2) = ²√2 = 2^(1/2) = ⁴√4 = 4^(1/4)

2 ✱ 1 = 2 = (²√2)² = ^(1/2)√(²√2) = ¹√2 = 2¹ = ²√4 = 4^(1/2)

2 ✱ 2 = 4 = (²√2)⁴ = ^(1/4)√(²√2) = ^(1/2)√2 = 2² = ¹√4 = 4¹

The pattern contunes. ²√2 is the square root of 2 and the 4th root of 4. 2 is ²√2 squared and the square root of 4. 4 is ²√2 to the exponent of 4 and 2 squared.

Therefore, 2 ✱ 0 = ²√2, or the value of 1.4142135623730950488016887242097

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Fun With Math-Superpowerition: Inventing a Super Power Operation Main Page Introduction Comparing The Math Operators I Comparing The Math Operators II Using Numbers 3 and Higher 1 ✱ b a ✱ 1 a ✱ 0 0 ✱ b a ✱ -b -a ✱ b -a ✱ -b Formula Summary So Far Integer a > 1 ✱ Decimal b > 1 Integer a > 1 ✱ Decimal 0 < b < 1 Integer a > 1 ✱ Decimal b < 0 Decimal a > 1 ✱ Decimal b > 1 Decimal a > 1 ✱ Decimal 0 < b < 1 Decimal a > 1 ✱ Decimal b < 0 Decimal 0 < a < 1 ✱ Decimal b > 1 Decimal 0 < a < 1 ✱ Decimal 0 < b < 1 Decimal 0 < a < 1 ✱ Decimal b < 0 Decimal -2 < a < 0 ✱ Decimal -1 < b < 2 Formula Summary So Far II Why 2 ✱ 0 is Not 1 Finding The Inverse Operations 1.25 ✱ -10 to 10 1.5 ✱ -10 to 10 1.75 ✱ -10 to 10 2 ✱ -10 to 10 2.5 ✱ -10 to 10 3 ✱ -10 to 10 3.5 ✱ -10 to 10 4 ✱ -10 to 10 4.5 ✱ -10 to 10 5 ✱ -10 to 10 6 ✱ -10 to 10 7 ✱ -10 to 10 8 ✱ -10 to 10 9 ✱ -10 to 10 10 ✱ -10 to 10
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