Powers, exponential notation

Similarly, the subtraction of the exponential notation may be performed by changing the expressions to forms having the same common power of 10 and then subtracting the coefficients. 

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. 

Addition or subtraction gets easier when you express your numbers as coefficients of identical powers of 10. To wrestle your numbers into this form, you may need to use coefficients less than 1 or greater than 10. So scientific notation is a bit too strict for addition and subtraction, but exponential notation still serves you well. 

To add two numbers easily by using exponential notation, first express each number as a coefficient and a power of 10, making sure that 10 is raised to the Scime exponent in each number. Then add the coefficients. To subtract numbers in exponential notation, follow the same steps but subtract the coefficients. 

This example involves concepts and units which may be unfamiliar to you, so that you can t easily judge whether the result "makes sense," so we ll check the answer by estimation. Write each term in exponential notation, using just one significant figure. Then mentally combine the powers of 10 and the multipliers separately to estimate the result ... 

Exponential notation enables easy reporting of extremely large and extremely small numbers. A number in scientific notation consists of a coefficient times 10 to an integral power, where the coefficient is equal to or greater than 1 but less than 10. Learn how to convert numbers from exponential notation to ordinary decimal values, and vice versa, and also how to use exponential numbers in calculations. Also learn to use effectively an electronic calculator with exponential capability (see Appendix 1). (Section 2.2)... 

To represent a number smaller than 1 in exponential notation, start with a number between 1 and 10 and divide by the appropriate power of 10 ... 

When a number expressed in exponential notation is taken to some power, the initial number is taken to the appropriate power and the exponent of 10 is multiplied by that power ... 

Exponential notation is an alternative way of expressing numbers in the form fl ( a to the power ), where a is multiplied by itself n times. The number a is called the base and the number n the exponent (or power or index). The exponent need not be a whole number, and it can be negative if the number being expressed is less than 1. See Table 39.2 for other mathematical relationships involving exponents. 

The SI uses seven base units, which are listed in Table B.l. All other units can be written as combinations of the base units. In writing the units for a measurement, we abbreviate them (see Table B.l), and we use exponential notation to denote the power to which a unit is raised a minus sign appears in the exponent... 

Raising to a Power a Number Written in Exponential Notation... 

Scientists report numbers from literally astronomical to almost infinitesimal. In order to do so conveniently, we use scientific notation, also known as standard exponential notation. Scientific notation is a form of a number with a decimal coefficient times a power of 10. The following number is in scientific notation, with its parts identified ... 

It is often difficult to express very large numbers to the proper number of significant figures using conventional notation. The solution to this problem lies in the use of scientific notation, also referred to as exponential notation, which involves the representation of a number as a power of ten. 

SI units consist of seven base units and numerous derived units. Exponential notation and prefixes based on powers of 10 are used to express very small and very large numbers. The SI base unit of length is the meter (m). Length units on the atomic scale are the nanometer (nm) and picometer (pm). Volume units are derived from length units the most important volume units are the cubic meter (m ) and the liter... 

Addition and Subtraction In order to add or subtract numbers expressed in exponential notation, the powers of 10 must be the same. 

Powers and Roots When numbers expressed in exponential notation are raised to a power, the exponents are multiplied by the power. When the roots of numbers expressed in exponential notation are taken, the exponents are divided by the root. 

The prefixes in Table 2.4 are represented by the powers of 10 used in scientific, or exponential, notation for writing large and small numbers. For example, 10 = 10 X 10 X 10 = 1000. Appendix B reviews this notation. 

Scientific notation, also known as exponential notation, is a way of representing large and small numbers as the product of two terms. The first term, the coefficient, is a number between 1 and 10. The second term, the exponential term, is 10 raised to a power—the exponent. For example. 

In exponential notation, a number is represented as a value raised to a power of ten. The decimal point can be located anywhere within the number as long as the power of ten is correct. In scientific notation, the decimal point is always located between the first and second digit — and the first digit must be a number other than zero. 

To multiply numbers expressed in exponential notation, multipiy the coefficients (the numbers) and add the exponents (powers of ten) ... 

To raise a number in exponential notation to a certain power, raise the coefficient to the power and then multiply the exponent by the power ... 

You can use a calculator to add and subtract numbers in exponential notation without first converting them to the same power of ten. The only thing you need to be careful about is entering the exponential number correctly. I m going to show you how to do that right now ... 

In scientific measurement and calculations, we often encounter very large and very small numbers—for example, 0.00000384 and 602,000,000,000,000,000,000,000. These numbers are troublesome to write and awkward to work with, especially in calculations. A convenient method of expressing these large and small numbers in a simplified form is by means of exponents, or powers, of 10. This method of expressing numbers is known as scientific, or exponential, notation. 

The exponent of 10 (4) tells us the number of places that the decimal point has been moved from its original position. If the decimal point is moved to the left, the exponent is a positive number if it is moved to the right, the exponent is a negative number. To express the number 0.00248 in exponential notation (as a power of 10), the decimal point is moved three places to the right the exponent of 10 is -3, and the number is 2.48 X 10 . ... 

A number in exponential notation consists of a digital number equal to or greater than exactly 1 and less than exactly 10 (e.g., 1.00000,4.3,6.913,8.005, and 9.99999) multiplied by a power of 10 (e.g., 10 , 10, 10 , 10, and Some examples of numbers expressed in exponential notation are given in Ihble 1.2. As can be seen in the second column of the table, a positive power of 10 shows the number of times that the digital number is multiplied by 10 and a negative power of 10 shows the number of times that the digital number is divided by 10. 

---