Math 6 Honors ( Periods 1, 2, & 3)

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000

10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620

762 X 10 = 7620

4931 X 10 = 49,310


10⁴ = 10⋅10⋅10⋅10 = 10,000

2.63874 X 10⁴ = 26,387.4

To multiply a number by the nth power of ten--> move the decimal n places to the right.

0.0047 multiply by 100 = 0.47
0.0047 multiply by 1000 = 4.7

3.1 ÷ 10⁴ = 0.00031


10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the n^th power of ten, move the decimal point n places to the right.



When we move a decimal point to the left, we are actually dividing by a power of ten.


Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

2386 ÷ 10³ = 2.386

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors

(1) a number greater than or equal to 1, but less than 10,

AND

(2) a power of ten

are said to be written in scientific notation.

We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation



4.592 x 10⁹ (Yes, you get to use the × symbol for multiplication .. but only for this!!



Way to write very large numbers AND very small numbers

Numbers expressed as products of a number greater than or equal to 1 BUT less than 10, AND a power of ten are called Scientific Notation.

Two Factors
91) 1≤ n < 10 (2) Power of 10 4,592,000,000 becomes 4.592 X 10⁹
moved the decimal 9 places so we must multiply our number by a power of 10⁹

98,000,000 = 9.8 X 10⁷

320,000 = 3.2 X 10⁵

What if I give you 7.04 X 10⁸ and ask you to put it back into STANDARD NOTATION:

704,000,000.

0.0031 = 3.1 X 10^-3
It isn't a negative number its just a very tiny number

1≤ n < 10 0.16 becomes 1.6 x 10 ^-1