Arithmetic with powers of ten

10 = 10 = 10¹

100 = 10 x 10 = 10²

1,000 = 10 x 10 x 10 = 10³

1,000,000=10 x 10 x 10 x 10 x 10 x 10 = 10⁶

MULTIPLYING TWO NUMBERS

1,000 x 1,000 = 1,000,000

It would be equivalent to saying that

10³ x 10³ = 10⁶.

Notice that the exponent "6" is just the sum of the other two exponents "3" and "3"! This, in fact is a basic rule whenever you multiply two numbers with the same "base" (in this case, 10). For ANY values a and b, it is always true that

10^a x 10^b = 10^a+b                       Example (10³x10⁵=10³⁺⁵=10⁸)

Now, the exponent can also be a negative number. In this way, we can write

Suppose a+b =0. What then? Let’s figure it out by taking a specific example:

10 x 1/10 = 10¹ x 10^-1

What is one-tenth of ten (or equivalently ten divided by ten)? Answer: 1

What number do you get when you add the two exponents (1 and —1)? Answer: 0

So then, what number is the same as 10⁰ = 10^1+(-1)? Answer:  10⁰ =1

100,000/1,000 = 100 — i.e., 10⁵/10³=10² = 10^5-3. This suggests that when dividing two numbers with the same base, the result can be found by subtracting the exponents:

10^a/10^b = 10^a-b.

Let’s look at an earlier example again:

1,000 x 1,000 = 1,000,000

It would be equivalent to saying that

10³ x 10³ = (10³)² = 10⁶ = 10^3x2

Thus, when you raise a number to a higher power, the final number can also be expressed in the same way, with the final exponent being equal to the original exponent TIMES that power.

(10^a)^b = 10^(axb)

The reverse process also works. If the square of 1,000 is 1,000,000, then the square root of 1,000,000 is 1,000. That is,

10³ = 10^6/2 = 10^6x(1/2) = (10⁶)^1/2

Most numbers are not simple multiples of 10, but they still can be expressed using powers-of-10 notation. I.e.,

26 = 2.6x10 = 2.6x10¹

45,221 = 4.5221x10,000 = 4.5221x10⁴

0.786 = 7.86x(1/10) = 7.86x10^-1

We generally write these as was done in this last case, a decimal number between 1 and 10 time the power of 10.

Likewise, we can multiply and divide numbers as long as we follow the rules we discovered before:

2x10³ x 4x 0³² = 2 x 4 x 10³ x 10³² = (2x4)x10³⁺³² = 8x10³⁵

Here, we used another rule of algebra you probably heard about in school: the "commutative property (or rule) of multiplication", a fancy was to say that it doesn’t matter what order you multiply two numbers in, the product is the same either way (2x4 = 4x2 = 8).

We can divide numbers:

(4x10³²)/(2x10³) = (4/2)x(10³²/10³) = 2x10²⁹

We can raise numbers to exponents:

(4x10³²)² = (4x10³²)x(4x10³²) = (4x4)x(10³²x10³²) = 16x10⁶⁴ = 1.6x10⁶⁵