10.5 Integer Exponents and Scientific Notation

Learning Objectives

By the end of this section, you will be able to:

  • Use the definition of a negative exponent
  • Simplify expressions with integer exponents
  • Convert from decimal notation to scientific notation
  • Convert scientific notation to decimal form
  • Multiply and divide using scientific notation

Use the Definition of a Negative Exponent

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.

Simplify Expressions with Integer Exponents

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.

Convert from Decimal Notation to Scientific Notation

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.On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, “Moved the decimal point 3 places to the left.” On the right, we see 0.004 equals 4 times 10 to the negative 3. Beneath that is the same thing, but there is an arrow from the decimal point to after the 4. Beneath, it says, “Moved the decimal point 3 places to the right.”

In both cases, the decimal was moved 3 places to get the first factor, 4, by itself.

  • The power of 10 is positive when the number is larger than 1:4000=4×103.
  • The power of 10 is negative when the number is between 0 and 1:0.004=4×10−3.

Convert Scientific Notation to Decimal Form

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.

On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200.  On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912.

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.

Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Media

ACCESS ADDITIONAL ONLINE RESOURCES

  • Negative Exponents
  • Examples of Simplifying Expressions with Negative Exponents
  • Scientific Notation

Section 10.5 Exercises

Practice Makes Perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Everyday Math

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.

  • ⓐ Write 1.5 trillion in decimal notation.
  • ⓑ Write 1.5 trillion in scientific notation.

417.Length of a year The difference between the calendar year and the astronomical year is 0.000125 day.

  • ⓐ Write this number in scientific notation.
  • ⓑ How many years does it take for the difference to become 1 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.

Writing Exercises

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?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?