By the end of this section, you will be able to:
The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.
What if we just subtract exponents, regardless of which is larger? Let’s consider x2/x5.
We subtract the exponent in the denominator from the exponent in the numerator.
The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.
When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.
We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results.
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.
All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.
If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
To simplify a fraction, we use the Quotient Property.
Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on.
Consider the numbers 4000 and 0.004. We know that 4000 means 4×1000 and 0.004 means 4×11000. If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:
When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.
It is customary in scientific notation to use × as the multiplication sign, even though we avoid using this sign elsewhere in algebra.
Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.
If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.
In both cases, the decimal was moved 3 places to get the first factor, 4, by itself.
How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.
If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.
In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.
We use the Properties of Exponents to multiply and divide numbers in scientific notation.
Use the Definition of a Negative Exponent
In the following exercises, simplify.
416.Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by 1.5 trillion calories by the end of 2015.
417.Length of a year The difference between the calendar year and the astronomical year is 0.000125 day.
418.Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided 1 by 2,598,960 and saw the answer 3.848×10−7. Write the number in decimal notation.419.
Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with 6 of her 50 favorite photographs, she multiplied 50⋅49⋅48⋅47⋅46⋅45. Her calculator gave the answer 1.1441304×1010. Write the number in decimal notation.
420.ⓐ Explain the meaning of the exponent in the expression 23.
ⓑ Explain the meaning of the exponent in the expression 2−3
421.When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?