Simplify: 2.5 x 10-¹² x 1.6 x 10¹⁶ x 1.3 x 10-²²
Answers
Answer:
Step-by-step explanation:
To write a large number in scientific notation, move the decimal point to the left to obtain a number between 1 and 10. Since moving the decimal point changes the value, you have to multiply the decimal by a power of 10 so that the expression has the same value.
Let’s look at an example.
180,000. = 18,000.0 x 101
1,800.00 x 102
180.000 x 103
18.0000 x 104
1.80000 x 105
180,000 = 1.8 x 105
Notice that the decimal point was moved 5 places to the left, and the exponent is 5.
The world population is estimated to be about 6,800,000,000 people. Which answer expresses this number in scientific notation?
A) 7 x 109
B) 0.68 x 1010
C) 6.8 x 109
D) 68 x 108
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Advanced Question
Represent 1.00357 x 10-6 in decimal form.
A) 1.00357000000
B) 0.000100357
C) 0.000001357
D) 0.00000100357
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To write a small number (between 0 and 1) in scientific notation, you move the decimal to the right and the exponent will have to be negative.
0.00004 = 00.0004 x 10-1
000.004 x 10-2
0000.04 x 10-3
00000.4 x 10-4
000004. x 10-5
0.00004 = 4 x 10-5
You may notice that the decimal point was moved five places to the right until you got the number 4, which is between 1 and 10. The exponent is −5.
Writing Scientific Notation in Decimal Notation
You can also write scientific notation as decimal notation. For example, the number of miles that light travels in a year is 5.88 x 1012, and a hydrogen atom has a diameter of 5 x 10-8 mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.
For each power of 10, you move the decimal point one place. Be careful here and don’t get carried away with the zeros—the number of zeros after the decimal point will always be 1 less than the exponent because it takes one power of 10 to shift that first number to the left of the decimal.
Rewrite 1.57 x 10-10 in decimal notation.
A) 15,700,000,000
B) 0.000000000157
C) 0.0000000000157
D) 157 x 10-12
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Multiplying and Dividing Numbers Expressed in Scientific Notation
Numbers that are written in scientific notation can be multiplied and divided rather simply by taking advantage of the properties of numbers and the rules of exponents that you may recall. To multiply numbers in scientific notation, first multiply the numbers that aren’t powers of 10 (the a in a x 10n). Then multiply the powers of ten by adding the exponents.
This will produce a new number times a different power of 10. All you have to do is check to make sure this new value is in scientific notation. If it isn’t, you convert it.
Let’s look at some examples.
Example
Problem
(3 x 108)(6.8 x 10-13)
(3 x 6.8)(108 x 10-13)
Regroup, using the commutative and associative properties.
(20.4)(108 x 10-13)
Multiply the coefficients.
20.4 x 10-5
Multiply the powers of 10, using the Product Rule—add the exponents.
(2.04 x 101) x 10-5
Convert 20.4 into scientific notation by moving the decimal point one place to the left and multiplying by 101.
2.04 x (101 x 10-5)
Group the powers of 10 using the associative property of multiplication.
2.04 x 101+(-5)
Multiply using the Product Rule—
add the exponents.
Answer
(3 x 108)(6.8 x 10-13) = 2.04 x 10-4
Advanced Example
Problem
(8.2 x 106)(1.5 x 10-3)(1.9 x 10-7)
Answer:
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