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0.05 In Scientific Notation

When functioning through very large or extremely small numbers, scientists, mathematicians, and designers often usage A positive number is created in scientific notation if it is composed as a x 10n where the coefficient a has actually a worth such that 1 ≤ a 10 and n is an integer.

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to expush those amounts. Scientific notation offers exponential notation. The following are examples of clinical notation.

Light year: number of miles light travels in one year, around 5,880,000,000,000

Scientific notation is 5.88 x 1012 miles.

hydrogen atom: has a diameter of about 0.00000005 mm

Scientific notation is 5 x 10-8 mm

Computation with exceptionally large numbers is made less complicated with clinical notation.


When a number is written in scientific notation, the When a number is expressed in the create ab, b is the exponent. The exponent shows how many times the base is supplied as a aspect. Power and exponent mean the exact same point.


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tells you if the term is a large or a small number. A positive exponent suggests a large number and also an adverse exponent shows a tiny number that is in between 0 and 1.

Due to the fact that it’s so useful, let’s look more closely at the details of scientific notation format.

Scientific Notation

A positive number is created in scientific notation if it is written as a x 10n where the coefficient a has actually a value such that 1 ≤ a n is an integer.

Look at the numbers below. Which of the numbers is created in scientific notation?


Number

Scientific Notation?

Explanation

1.85 x 10-2

yes

1 ≤1.85

-2 is an integer

*

no

*
 is not an integer

0.82 x 1014

no

0.82 is not ≥ 1

10 x 103

no

10 is not


Which number below is created in scientific notation?

A) 4.25 x 100.08

B) 0.425 x 107

C) 42.5 x 105

D) 4.25 x 106


Show/Hide Answer

A) 4.25 x 100.08

Incorrect. The exponent should be an integer and also 0.08 is not an integer. The correct answer is 4.25 x 106.

B) 0.425 x 107

Incorrect. This is not in clinical notation bereason 0.425 is less than 1. The correct answer is 4.25 x 106.

C) 42.5 x 105

Incorrect. This is not in scientific notation because 42.5 is higher than 10. The correct answer is 4.25 x 106.

D) 4.25 x 106

Correct. This is scientific notation. 4.25 is better than 1 and also less than 10, and 6 is an integer.

Writing Decimal Notation in Scientific Notation


Now let’s compare some numbers expressed in both scientific notation and traditional decimal notation in order to understand also how to convert from one form to the various other. Take a look at the tables below. Pay cshed attention to the exponent in the scientific notation and the position of the decimal suggest in the decimal notation.


Large Numbers

Small Numbers

Decimal Notation

Scientific Notation

Decimal Notation

Scientific Notation

500.0

5 x 102

0.05

5 x 10-2

80,000.0

8 x 104

0.0008

8 x 10-4

43,000,000.0

4.3 x 107

0.00000043

4.3 x 10-7

62,500,000,000.0

6.25 x 1010

0.000000000625

6.25 x 10-10


To create a big number in clinical notation, relocate the decimal allude to the left to achieve a number between 1 and also 10. Since moving the decimal allude transforms the worth, you have to multiply the decimal by a power of 10 so that the expression has the same worth.

Let’s look at an instance.

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 relocated 5 areas to the left, and the exponent is 5.

The people population is estimated to be around 6,800,000,000 human being. Which answer expresses this number in clinical notation?

A) 7 x 109

B) 0.68 x 1010

C) 6.8 x 109

D) 68 x 108


Show/Hide Answer

A) 7 x 109

Incorrect. Scientific notation rewrites numbers, it doesn’t round them. The correct answer is 6.8 x 109.

B) 0.68 x 1010

Incorrect. Although 0.68 x 1010 is equivalent to 6,800,000,000. 0.68 is not the develop for clinical notation as 0.68 is not a number between 1 and also 10. The correct answer is 6.8 x 109.

C) 6.8 x 109

Correct. The number 6.8 x 109 is tantamount to 6,800,000,000 and also provides the proper format for each variable.

D) 68 x 108

Incorrect. Although 68 x 108 is indistinguishable to 6,800,000,000, it is not written in scientific notation as 68 is not between 1 and 10. The correct answer is 6.8 x 109.

State-of-the-art Question

Reexisting 1.00357 x 10-6 in decimal form.

A) 1.00357000000

B) 0.000100357

C) 0.000001357

D) 0.00000100357


Show/Hide Answer

A) 1.00357000000

B) 0.000100357

Incorrect. You moved the decimal allude in the correct direction, however you did not move it sufficient areas. The correct answer is 0.00000100357.

C) 0.000001357

Incorrect. You relocated the decimal allude the correct number of spaces, yet the number you produced is different than the number you began with: 1.00357 x 10-6 ≠ 0.000001357. Remember that the zeroes in in between 1 and 3 need to additionally be consisted of in the last number. The correct answer is 0.00000100357.

D) 0.00000100357

Correct. The exponent is -6. You moved the decimal allude 6 spots to the left, developing the decimal 0.00000100357.

To create a small number (in between 0 and 1) in clinical notation, you move the decimal to the best and also 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 allude was relocated five areas to the right till you acquired the number 4, which is in between 1 and also 10. The exponent is −5.


Writing Scientific Notation in Decimal Notation


You have the right to likewise write clinical 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 actually a diameter of 5 x 10-8 mm. To write each of these numbers in decimal notation, you move the decimal point the very same number of locations as the exponent. If the exponent is positive, relocate the decimal allude to the best. If the exponent is negative, move the decimal point to the left.

*

For each power of 10, you relocate the decimal allude one area. Be mindful right here and also don’t gain carried ameans with the zeros—the variety of zeros after the decimal point will always be 1 much less than the exponent bereason it takes one power of 10 to transition that initially 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


Show/Hide Answer

A) 15,700,000,000

Incorrect. You moved the decimal suggest in the wrong direction. The exponent is negative, so to transform to decimal format move the decimal suggest to the left, not the right. The correct answer is 0.000000000157.

B) 0.000000000157

Correct. The expression has actually a negative exponent, so you relocate the decimal point 10 places to the left to transform it to decimal notation, 0.000000000157.

C) 0.0000000000157

Incorrect. You inserted 10 zeros between the number and the decimal point. Move the decimal point 10 places to the left instead. The correct answer is 0.000000000157.

D) 157 x 10-12

Incorrect. This number is identical to the original number, yet it is not in decimal notation. The correct answer is 0.000000000157.

Multiplying and Dividing Numbers Expressed in Scientific Notation


Numbers that are written in scientific notation deserve to be multiplied and also split rather sindicate by taking benefit of the properties of numbers and the rules of exponents that you may respeak to. 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 including the exponents.

This will produce a new number times a different power of 10. All you need to do is examine to make certain this new worth is in clinical notation. If it isn’t, you transform 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)

Reteam, using the commutative and also associative properties.

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(20.4)(108 x 10-13)

Multiply the coefficients.

20.4 x 10-5

Multiply the powers of 10, making use of the Product Rule—include the exponents.

(2.04 x 101) x 10-5

Convert 20.4 into clinical notation by relocating the decimal allude one area to the left and multiplying by 101.

2.04 x (101 x 10-5)

Group the powers of 10 making use of the associative home of multiplication.

2.04 x 101+(-5)

Multiply making use of the Product Rule—

 add the exponents.

Answer

(3 x 108)(6.8 x 10-13) = 2.04 x 10-4


Cutting edge Example

Problem

(8.2 x 106)(1.5 x 10-3)(1.9 x 10-7)

(8.2 x 1.5 x 1.9)(106 x 10-3 x 10-7)

Regroup, using the commutative and also associative properties.

(23.37) (106 x 10-3 x 10-7)

Multiply the numbers.

23.37 x 10-4

Multiply the powers of 10, utilizing the Product Rule—include the exponents.

(2.337 x 101) x 10-4

Convert 23.37 right into scientific notation by moving the decimal point one area to the left and also multiplying by 101.

2.337 x (101 x 10-4)

Group the powers of 10 utilizing the associative property of multiplication.

2.337 x 101+(-4)

Multiply making use of the Product Rule and add the exponents.

Answer

(8.2 x 106)(1.5 x 10-3)(1.9 x 10-7) = 2.337 x 10-3


In order to divide numbers in scientific notation, you when aobtain use the properties of numbers and also the rules of exponents. You begin by dividing the numbers that aren’t powers of 10 (the a in a x 10n). Then you divide the powers of ten by subtracting the exponents.

This will develop a new number times a various power of 10. If it isn’t already in scientific notation, you convert it, and then you’re done.

Let’s look at some examples.


Example

Problem

*

*

Regroup, making use of the associative building.

(0.82)

*

Divide the coefficients.

0.82 x 10-9 – (-3)

0.82 x 10-6

Divide the powers of 10 utilizing the Quotient Rule—subtract the exponents.

(8.2 ´ 10-1) x 10-6

Convert 0.82 into scientific notation by moving the decimal allude one location to the ideal and multiplying by 10-1.

8.2 ´ (10-1 x 10-6)

Group the powers of 10 together utilizing the associative residential property.

8.2 x 10-1+(-6)

Multiply the powers of 10, utilizing the Product Rule—include the exponents.

Answer

*


Advanced Example

Problem

*

*

Regroup the terms in the numerator according to the associative and also commutative properties.

*

Multiply.

*

Reteam, utilizing the associative residential property.

*

Divide the numbers.

*

*

Divide the powers of 10 making use of the Quotient Rule—subtract the exponents.

Answer

*


Notice that as soon as you divide exponential terms, you subtract the exponent in the denominator from the exponent in the numerator.

Evaluate (4 x 10-10)(3 x 105) and also expush the bring about clinical notation.

A) 1.2 x 10-4

B) 12 x 10-5

C) 7 x 10-5

D) 1.2 x 10-50


Show/Hide Answer

A) 1.2 x 10-4

Correct. 1.2 x 10-4 is an exact computation and correct scientific notation.

B) 12 x 10-5

Incorrect. Ala lot of correct, but now you have to transform the coeffective 12 into clinical notation. 12 is better than 10 and scientific notation requires this number to be greater than or equal to 1 and less than 10. The correct answer is 1.2 x 10-4.

C) 7 x 10-5

Incorrect. Multiply, not add, the numbers 4 and 3. The correct answer is 1.2 x 10-4.

D) 1.2 x 10-50

Incorrect. Add, not multiply, exponents. The correct answer is 1.2 x 10-4.

Modern Question

Evaluate (3.15 x 104)(5.15 x 10-7) and expush the outcome as a decimal.

A) 0.0162225

B) 162225

C) 0.000162225

D) 16.2225


Show/Hide Answer

A) 0.0162225

Correct. 3.15 x 5.15 = 16.2225, and also 104 x 10-7 = 10-3. The exponent is negative, so to convert to decimal format relocate the decimal point 3 spaces to the left for a worth of 0.0162225.

B) 162225

Incorrect. The expression has an unfavorable exponent, so you move the decimal suggest 3 places to the left, not to the appropriate, to convert it to decimal notation. The correct answer is 0.0162225.

C) 0.000162225

Incorrect. You relocated the decimal point in the correct direction, yet not the correct variety of spots. Remember that 104 x 10-7 = 10-3, so you need to relocate the decimal suggest three spots to the left. The correct answer is 0.0162225.

D) 16.2225

Incorrect. 3.15 x 5.15 = 16.2225, but it looks favor you forobtained to multiply 104 x 10-7. The correct answer is 0.0162225.

Summary


Scientific notation was arisen to help mathematicians, scientists, and others once expressing and also working through very huge and incredibly small numbers. Scientific notation complies with a really particular format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and also a power of 10. The format is composed a x 10n, where 1 ≤ a n is an integer.

To multiply or divide numbers in scientific notation, you have the right to usage the commutative and also associative properties to group the exponential terms together and use the rules of exponents.