Exponents, scientific notation

Scientific notation uses exponents to express numbers. The number 1,000, for instance, is equal to 10 x 10 x 10, or 10. The number of zeros following the 1 in 1,000 is 3, the same as the exponent in scientific notation. Similarly, 10,000, with 4 zeros, would be 10 , and so on. The same rules apply to numbers that are not even multiples of 10. For example, the number 1,360 is 1.36 x 10. And the number of atoms in a spoonful of water becomes an easy-to-write 5 X 10. ... 

The abbreviation log stands for logarithm. In mathematics, a logarithm is the power (also called an exponent) to which a number (called the base) has to be raised to get a particular number. In other words, it is the number of times the base (this is the mathematical base, not a chemical base) must be multiplied times itself to get a particular number. For example, if the base number is 10 and 1,000 is the number trying to be reached, the logarithm is 3 because 10 x 10 x 10 equals 1,000. Another way to look at this is to put the number 1,000 into scientific notation ... 

Note (2) It is more difficult to find K in scientific notation because most calculators cannot handle numbers this big. So, use what you know about exponents to solve for K ... 

You can use these laws of exponents with scientific notation. 

Click on any of these topics Exponents, Radicals, or Scientific Notation. 

Rule 1 To multiply two numbers in scientific notation, add the exponents. 

Rule 3 To add or subtract numbers in scientific notation, first convert the numbers so they have the same exponent. Each number should have the same exponent as the number with the greatest power of 10. Once the numbers are all expressed to the same power of 10, the power of 10 is neither added nor subtracted in the calculation. 

To make working with such extreme numbers easier, chemists turn to scientific notation, which is a special kind of exponential notation. Exponential notation simply means writing a number in a way that includes exponents. In scientific notation, every number is written as the product of two numbers, a coefficient and a power of 10. In plain old exponential notation, a coefficient can be any value of a number multiplied by a power with a base of 10 (such as 10" ). But scientists have rules for coefficients in scientific notation. In scientific notation, the coefficient is always at least 1 and always less than 10. For example, the coefficient could be 7, 3.48, or 6.0001. 

To multiply two numbers written in scientific notation, multiply the coefficients and then add the exponents. To divide two numbers, simply divide the coefficients and then subtract the exponent of the denominator (the bottom number) from the exponent of the numerator (the top number). 

X10 The ease of math with scientific notation shines through in this problem. Dividing the coefficients yields a coefficient quotient of 9.3/3.1 =3.0, and dividing the powers of ten (by subtracting their exponents) yields a quotient of 10 /10 = 10" " = 10 . Marrying the two quotients produces the given answer, already in scientific notation. 

X10" First, convert each number to scientific notation 8.09x 10 and 2.03X10. Then divide the coefficients 8.09/2.03 = 3.99. Next, subtract the exponent on the denominator from the exponent of the numerator to get the new power of 10 10 = 10" Join the new coefficient with the new power 3.99x10 . Finally, express gratitude that the answer is already conveniently expressed in scientific notation. 

In the addition or subtraction of numbers expressed in scientific notation, all numbers should first be expressed with the same exponent ... 

To understand how this shorthand notation works, consider the large number 50,000,000. Mathematically this number is equal to 5 multiplied by 10 X 10X 10X 10X 10 X 10 X 10 (check this out on your calculator). We can abbreviate this chain of numbers by writing all the 10s in exponential form, which gives us the scientific notation 5 X 107. (Note that 107 is the same as lOx lOx 10x lOx 10 X 10 X 10. Table A. 1 shows the exponential form of some other large and small numbers.) Likewise, the small number 0.0005 is mathematically equal to 5 divided by 10 X 10 x 10 X 10, which is 5/104. Because dividing by a number is exactly equivalent to multiplying by the reciprocal of that number, 5/104 can be written in the form 5 X 10-4, and so in scientific notation 0.0005 becomes 5 X 10-4 (note the negative exponent). 

Because we are thus moving the decimal point four places to the right we must decrease the exponent on the 10° by 4 Therefore the scientific notation is 5 5 x 10"4... 

To use the scientific notation of numbers in mathematical operations, we must remember the laws of exponents. 

To multiply two numbers, put them both in standard scientific notation. Then multiply the two lefthand factors by ordinary multiplication, and multiply the two righthand factors (powers of 10) by the multiplication law for exponents — that is, by adding their exponents. 

To divide one number by another, put them both in standard scientific notation. Divide the first lefthand factor by the second, according to the rules of ordinary division. Divide the first righthand factor by the second, according to the division law for exponents—that is, by subtracting the exponent of the divisor from the exponent of the dividend to obtain the exponent of the quotient. 

To add or subtract numbers in scientific notation, adjust the numbers to make all the exponents on the righthand factors the same. Then add or subtract the lefthand factors by the ordinary rules, making no further change in the righthand factors. 

Notice how numbers that are either very large or very small are indicated in Table 1.4 using an exponential format called scientific notation. For example, the number 55,000 is written in scientific notation as 5.5 X 104, and the number 0.003 20 as 3.20 X 10 3. Review Appendix A if you are uncomfortable with scientific notation or if you need to brush up on how to do mathematical manipulations on numbers with exponents. 

Scientific notation uses exponents (powers of 10) for handling very large or very small numbers. A number in scientific notation consists of a number multiplied by a power of 10. The number is called the mantissa. In scientific notation, only one digit in the mantissa is to the left of the decimal place. The order of magnitude is expressed as a power of 10, and indicates how many places you had to move the decimal point so that only one digit remains to the left of the decimal point. 

Numbers in proper scientific notation have only one digit in front of the decimal place. Can you convert 12.5 x 10 3 into proper form You have to move the decimal point one place to the left, but does the exponent increase or decrease Is the correct answer 1.25 x 10-4 or 1.25 x 10-2 Rather than memorize a rule, think of it this way. You want your answer to be equal to the initial number. So, if you DECREASE 12.5 to 1.25, wouldn t you need to INCREASE the exponent to cancel out this change So, the correct answer is 1.25 x 10-2. 

A couple of comments are in order here. First, did you notice the value of the pH is the same as the absolute value of the exponent This will always be true when the first part of the scientific notation is exactly 1. The second comment relates to significant figures. There are two significant figures in the molarity measurement of 1.0 x 10 3 M. There are also two significant figures in the pH value of 3.00. Finally, pH values have no intrinsic units. Logarithms represent pure numbers, and as such, have no units. 

In this unit you will find explanations, examples, and practice dealing with the calculations encountered in the chemistry discussed in this book. The types of calculations included here involve conversion factors, metric use, algebraic manipulations, scientific notation, and significant figures. This unit can be used by itself or be incorporated for assistance with individual units. Unless otherwise noted, all answers are rounded to the hundredth place. The calculator used here is a Casio FX-260. Any calculator that has a log (logarithm) key and an exp (exponent) key is sufficient for these chemical calculations. 

The term order of magnitude refers to the size of a number—specifically to its exponent when in scientific notation. For example, a scientist will say that 50 000 (5 x 104) is two orders of magnitude (102 times) larger than 500 (5 x 102). How many orders of magnitude is the Avogadro constant greater than one billion ... 

Numbers less than 1 are converted into scientific notation by moving the decimal point to the right of the left-most non-zero digit. The exponent is expressed as a negative power of 10. 

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