Scientific notation multiplication

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. ... 

A major benefit of presenting numbers in scientific notation is that it simplifies common arithmetic operations. The simplifying abilities of scientific notation cire most evident in multiplication and division. (As we note in the next section, addition and subtraction benefit from exponential notation but not necesscirily from strict scientific notation.)... 

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. 

The conventional representation of numbers using scientific notation is not always followed. For example, we might represent a bond length as 0.14x 10-9 m rather than 1.4 x 10-10 m if we were referencing it to another measurement of length given in integer unit multiples of 10 9 m. 

Time The SI base unit for time is the second (s). The frequency of microwave radiation given off by a cesium-133 atom is the physical standard used to establish the length of a second. Cesium clocks are more reliable than the clocks and stopwatches that you use to measure time. For ordinary tasks, a second is a short amount of time. Many chemical reactions take place in less than a second. To better describe the range of possible measurements, scientists add prefixes to the base units. This task is made easier because the metric system is a decimal system. The prefixes in Table 2-2 are based on multiples, or factors, of ten. These prefixes can be used with all SI units. In Section 2.2, you will learn to express quantities such as 0.000 000 015 s in scientific notation, which also is based on multiples of ten. 

Scientific notation expresses numbers as a multiple of two factors a number between 1 and 10 and ten raised to a power, or exponent. The exponent tells you how many times the first factor must be multiplied by ten. The mass of a proton is 1.627 62 X 10 kg in scientific notation. The mass of an electron is 9.109 39 X 10 kg. When numbers larger than 1 are expressed in scientific notation, the power of ten is positive. When numbers smaller than 1 are expressed in scientific notation, the power of ten is negative. 

Multiplying and dividing using scientific notation Multiplying and dividing also involve two steps, but in these cases the quantities being multiplied or divided do not have to have the same exponent. For multiplication, you multiply the first factors. Then, you add the exponents. For division, you divide the first factors. Then, you subtract the exponent of the divisor from the exponent of the dividend. 

Solve the following multiplication and division problems. Express your answers in scientific notation. 

When two or more numbers written in scientific notation are to be added or subtracted, they should first be expressed as multiples of the same power of 10 ... 

Scientists and engineers are frequently called upon to use very large or very small numbers for which ordinary decimal notation is either awkward or impossible. For example, to express Avogadro s number in decimal notation would require 21 zeros following the number 602. In scientific notation the number is written as a multiple of two numbers, the one number in decimal notation and the other expressed as a power of 10. Thus, Avogadro s number is written as 6.02 X 1Other examples are... 

Look at Fig re 7 to see how to solve multiplication and division problems involving scientific notation. The problems are the same as the two previous sample problems. 

You are accustomed to writing numbers in decimal notation, for example 123 677.54 and 0-001678. in working with large numbers and small numbers, you will fmd it convenient to write them in a (Afferent way, known as scientific notation oc stanJerd form. This means writing a number as a product of two factors. In the first factor, the Mimai point comes after the first digit. The second factor is a multiple of ten. For example, 2123 = 2.123 X 10 and 0.000167 = 1.67 X 10. lO means lOX lOX 10, and 10 means 1/(10 X 10 X 10 X 10). The number 3 or —4 is cdled the exponent, and the number 10 is the base. KH is referred ro as 10 to the power or 10 CO the third power . You will have noticed that, if the exponent it increased by 1, the decimal point must be moved one place to the left. 

Multiplication and Division To multiply numbers expressed in scientific notation, we multiply Ni and N2 in the usual way, but add the exponents together. To divide using scientific notation, we divide and N2 as usual and subtract the exponents. 

The multiplication and division of numbers written in scientific notation can be done qnite simply by using some characteristics of exponentials. Consider the following multipUcation ... 

Multiplication and division calculations involving scientific notation are easily done using a hand calculator, t Table 1.4 gives the steps, the typical calculator procedures (buttons to press), and typical calculator readout or display for the division of 7.2 X 10 by 1.2 X 10 . 

Do the following multiplications, and express each answer using scientific notation ... 

Recall that zeros have special rules and may require a contextual interpretation. As a starting point, a number may be converted to scientific notation. If the zeros can be removed by this operation, then they were merely placeholders representing a multiplication or division by 10. For example, suppose an instrument produces a result of 0.001023 that can be expressed as 1.023 X 10. This demonstrates that the leading zeros are not significant, but the embedded zero is. 

Multiplication and division in scientific notation when numbers expressed in scientific notation are being multipUed, the following general rule is very useful ... 

Numbers in scientific notation are entered in a fairly obvious way. To enter 1.234 X 10", you would type 1.234 lOM with the space standing for multiplication, or 1.234 lOM. In the output lines, Mathematica always uses the space for multiplication. 

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