10.6: 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

BE PREPARED 10.12

Before you get started, take this readiness quiz.

What is the place value of the 66 in the number 64,891?64,891?
If you missed this problem, review Example 1.3.

BE PREPARED 10.13

Name the decimal 0.0012.0.0012.
If you missed this problem, review [link].

BE PREPARED 10.14

Subtract: 5−(−3).5−(−3).
If you missed this problem, review Example 3.37.

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.

QUOTIENT PROPERTY OF EXPONENTS

If aa is a real number, a≠0,a≠0, and m,nm,n are whole numbers, then

aman=am−n,m>nandaman=1an−m,n>maman=am−n,m>nandaman=1an−m,n>m

What if we just subtract exponents, regardless of which is larger? Let’s consider x2x5.x2x5.

We subtract the exponent in the denominator from the exponent in the numerator.

x2x5x2x5

x2−5x2−5

x−3x−3

We can also simplify x2x5x2x5 by dividing out common factors: x2x5.x2x5.

This implies that x−3=1x3x−3=1x3 and it leads us to the definition of a negative exponent.

NEGATIVE EXPONENT

If nn is a positive integer and a≠0,a≠0, then a−n=1an.a−n=1an.

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.

EXAMPLE 10.63

Simplify:

1. ⓐ4−24−2
2. ⓑ10−310−3

Answer

TRY IT 10.125

Simplify:

1. ⓐ2−32−3
2. ⓑ10−210−2

TRY IT 10.126

Simplify:

1. ⓐ3−23−2
2. ⓑ10−410−4

When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.

EXAMPLE 10.64

Simplify:

1. ⓐ(−3)−2(−3)−2
2. ⓑ−3−2−3−2

Answer

TRY IT 10.127

Simplify:

1. ⓐ(−5)−2(−5)−2
2. ⓑ−5−2−5−2

TRY IT 10.128

Simplify:

1. ⓐ(−2)−2(−2)−2
2. ⓑ−2−2−2−2

We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results.

EXAMPLE 10.65

Simplify:

1. ⓐ4⋅2−14·2−1
2. ⓑ(4⋅2)−1(4·2)−1

Answer

TRY IT 10.129

Simplify:

1. ⓐ6⋅3−16·3−1
2. ⓑ(6⋅3)−1(6·3)−1

TRY IT 10.130

Simplify:

1. ⓐ8⋅2−28·2−2
2. ⓑ(8⋅2)−2(8·2)−2

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.

EXAMPLE 10.66

Simplify: x−6.x−6.

Answer

TRY IT 10.131

Simplify: y−7.y−7.

TRY IT 10.132

Simplify: z−8.z−8.

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.

EXAMPLE 10.67

Simplify:

1. ⓐ5y−15y−1
2. ⓑ(5y)−1(5y)−1
3. ⓒ(−5y)−1(−5y)−1

Answer

TRY IT 10.133

Simplify:

1. ⓐ8p−18p−1
2. ⓑ(8p)−1(8p)−1
3. ⓒ(−8p)−1(−8p)−1

TRY IT 10.134

Simplify:

1. ⓐ11q−111q−1
2. ⓑ(11q)−1(11q)−1
3. ⓒ(−11q)−1(−11q)−1

Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, aman=am−n,aman=am−n, where a≠0a≠0 and m and n are integers.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, a−n=1an.a−n=1an.

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.

SUMMARY OF EXPONENT PROPERTIES

If a,ba,b are real numbers and m,nm,n are integers, then

Product PropertyPower PropertyProduct to a Power PropertyQuotient PropertyZero Exponent PropertyQuotient to a Power PropertyDefinition of Negative Exponentam⋅an=am+n(am)n=am⋅n(ab)m=ambmaman=am−n,a≠0a0=1,a≠0(ab)m=ambm,b≠0a−n=1anProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambmQuotient Propertyaman=am−n,a≠0Zero Exponent Propertya0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0Definition of Negative Exponenta−n=1an

EXAMPLE 10.68

Simplify:

1. ⓐx−4⋅x6x−4·x6
2. ⓑy−6⋅y4y−6·y4
3. ⓒz−5⋅z−3z−5·z−3

Answer

TRY IT 10.135

Simplify:

1. ⓐx−3⋅x7x−3·x7
2. ⓑy−7⋅y2y−7·y2
3. ⓒz−4⋅z−5z−4·z−5

TRY IT 10.136

Simplify:

1. ⓐa−1⋅a6a−1·a6
2. ⓑb−8⋅b4b−8·b4
3. ⓒc−8⋅c−7c−8·c−7

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.

EXAMPLE 10.69

Simplify: (m4n−3)(m−5n−2).(m4n−3)(m−5n−2).

Answer

TRY IT 10.137

Simplify: (p6q−2)(p−9q−1).(p6q−2)(p−9q−1).

TRY IT 10.138

Simplify: (r5s−3)(r−7s−5).(r5s−3)(r−7s−5).

If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.

EXAMPLE 10.70

Simplify: (2x−6y8)(−5x5y−3).(2x−6y8)(−5x5y−3).

Answer

TRY IT 10.139

Simplify: (3u−5v7)(−4u4v−2).(3u−5v7)(−4u4v−2).

TRY IT 10.140

Simplify: (−6c−6d4)(−5c−2d−1).(−6c−6d4)(−5c−2d−1).

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

EXAMPLE 10.71

Simplify: (k3)−2.(k3)−2.

Answer

TRY IT 10.141

Simplify: (x4)−1.(x4)−1.

TRY IT 10.142

Simplify: (y2)−2.(y2)−2.

EXAMPLE 10.72

Simplify: (5x−3)2.(5x−3)2.

Answer

TRY IT 10.143

Simplify: (8a−4)2.(8a−4)2.

TRY IT 10.144

Simplify: (2c−4)3.(2c−4)3.

To simplify a fraction, we use the Quotient Property.

EXAMPLE 10.73

Simplify: r5r−4.r5r−4.

Answer

TRY IT 10.145

Simplify: x8x−3.x8x−3.

TRY IT 10.146

Simplify: y7y−6.y7y−6.

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 10.10. 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 40004000 and 0.004.0.004. We know that 40004000 means 4×10004×1000 and 0.0040.004 means 4×11000.4×11000. If we write the 10001000 as a power of ten in exponential form, we can rewrite these numbers in this way:

40004×10004×1030.0044×110004×11034×10−340000.0044×10004×110004×1034×11034×10−3

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 10,10, and the second factor is a power of 1010 written in exponential form, it is said to be in scientific notation.

SCIENTIFIC NOTATION

A number is expressed in scientific notation when it is of the form

a×10na×10n

where a≥1a≥1 and a<10a<10 and nn is an integer.

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 33 places to get the first factor, 4,4, by itself.

• The power of 1010 is positive when the number is larger than 1:4000=4×103.1:4000=4×103.
• The power of 1010 is negative when the number is between 00 and 1:0.004=4×10−3.1:0.004=4×10−3.

EXAMPLE 10.74

Write 37,00037,000 in scientific notation.

Answer

TRY IT 10.147

Write in scientific notation: 96,000.96,000.

TRY IT 10.148

Write in scientific notation: 48,300.48,300.

HOW TO

Convert from decimal notation to scientific notation.

1. Step 1. Move the decimal point so that the first factor is greater than or equal to 11 but less than 10.10.
2. Step 2. Count the number of decimal places, n,n, that the decimal point was moved.
3. Step 3. Write the number as a product with a power of 10.10.
   • If the original number is:
     • greater than 1,1, the power of 1010 will be 10n.10n.
     • between 00 and 1,1, the power of 1010 will be 10−n.10−n.
4. Step 4. Check.

EXAMPLE 10.75

Write in scientific notation: 0.0052.0.0052.

Answer

TRY IT 10.149

Write in scientific notation: 0.0078.0.0078.

TRY IT 10.150

Write in scientific notation: 0.0129.0.0129.

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.

9.12×1049.12×10,00091,2009.12×10−49.12×0.00010.0009129.12×1049.12×10−49.12×10,0009.12×0.000191,2000.000912

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.

EXAMPLE 10.76

Convert to decimal form: 6.2×103.6.2×103.

Answer

TRY IT 10.151

Convert to decimal form: 1.3×103.1.3×103.

TRY IT 10.152

Convert to decimal form: 9.25×104.9.25×104.

HOW TO

Convert scientific notation to decimal form.

1. Step 1. Determine the exponent, n,n, on the factor 10.10.
2. Step 2. Move the decimal nn places, adding zeros if needed.
   • If the exponent is positive, move the decimal point nn places to the right.
   • If the exponent is negative, move the decimal point |n||n| places to the left.
3. Step 3. Check.

EXAMPLE 10.77

Convert to decimal form: 8.9×10−2.8.9×10−2.

Answer

TRY IT 10.153

Convert to decimal form: 1.2×10−4.1.2×10−4.

TRY IT 10.154

Convert to decimal form: 7.5×10−2.7.5×10−2.

Multiply and Divide Using Scientific Notation

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

EXAMPLE 10.78

Multiply. Write answers in decimal form: (4×105)(2×10−7).(4×105)(2×10−7).

Answer

TRY IT 10.155

Multiply. Write answers in decimal form: (3×106)(2×10−8).(3×106)(2×10−8).

TRY IT 10.156

Multiply. Write answers in decimal form: (3×10−2)(3×10−1).(3×10−2)(3×10−1).

EXAMPLE 10.79

Divide. Write answers in decimal form: 9×1033×10−2.9×1033×10−2.

Answer

TRY IT 10.157

Divide. Write answers in decimal form: 8×1042×10−1.8×1042×10−1.

TRY IT 10.158

Divide. Write answers in decimal form: 8×1024×10−2.8×1024×10−2.

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.

316.

5−35−3

317.

8−28−2

318.

3−43−4

319.

2−52−5

320.

7−17−1

321.

10−110−1

322.

2−3+2−22−3+2−2

323.

3−2+3−13−2+3−1

324.

3−1+4−13−1+4−1

325.

10−1+2−110−1+2−1

326.

100−10−1+10−2100−10−1+10−2

327.

20−2−1+2−220−2−1+2−2

328.

1. ⓐ (−6)−2(−6)−2
2. ⓑ −6−2−6−2

329.

1. ⓐ (−8)−2(−8)−2
2. ⓑ −8−2−8−2

330.

1. ⓐ (−10)−4(−10)−4
2. ⓑ −10−4−10−4

331.

1. ⓐ (−4)−6(−4)−6
2. ⓑ −4−6−4−6

332.

1. ⓐ 5⋅2−15·2−1
2. ⓑ (5⋅2)−1(5·2)−1

333.

1. ⓐ 10⋅3−110·3−1
2. ⓑ (10⋅3)−1(10·3)−1

334.

1. ⓐ 4⋅10−34·10−3
2. ⓑ (4⋅10)−3(4·10)−3

335.

1. ⓐ 3⋅5−23·5−2
2. ⓑ (3⋅5)−2(3·5)−2

336.

n−4n−4

337.

p−3p−3

338.

c−10c−10

339.

m−5m−5

340.

1. ⓐ 4x−14x−1
2. ⓑ (4x)−1(4x)−1
3. ⓒ (−4x)−1(−4x)−1

341.

1. ⓐ 3q−13q−1
2. ⓑ (3q)−1(3q)−1
3. ⓒ (−3q)−1(−3q)−1

342.

1. ⓐ 6m−16m−1
2. ⓑ (6m)−1(6m)−1
3. ⓒ (−6m)−1(−6m)−1

343.

1. ⓐ 10k−110k−1
2. ⓑ (10k)−1(10k)−1
3. ⓒ (−10k)−1(−10k)−1

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

344.

p−4⋅p8p−4·p8

345.

r−2⋅r5r−2·r5

346.

n−10⋅n2n−10·n2

347.

q−8⋅q3q−8·q3

348.

k−3⋅k−2k−3·k−2

349.

z−6⋅z−2z−6·z−2

350.

a⋅a−4a·a−4

351.

m⋅m−2m·m−2

352.

p5⋅p−2⋅p−4p5·p−2·p−4

353.

x4⋅x−2⋅x−3x4·x−2·x−3

354.

a3b−3a3b−3

355.

u2v−2u2v−2

356.

(x5y−1)(x−10y−3)(x5y−1)(x−10y−3)

357.

(a3b−3)(a−5b−1)(a3b−3)(a−5b−1)

358.

(uv−2)(u−5v−4)(uv−2)(u−5v−4)

359.

(pq−4)(p−6q−3)(pq−4)(p−6q−3)

360.

(−2r−3s9)(6r4s−5)(−2r−3s9)(6r4s−5)

361.

(−3p−5q8)(7p2q−3)(−3p−5q8)(7p2q−3)

362.

(−6m−8n−5)(−9m4n2)(−6m−8n−5)(−9m4n2)

363.

(−8a−5b−4)(−4a2b3)(−8a−5b−4)(−4a2b3)

364.

(a3)−3(a3)−3

365.

(q10)−10(q10)−10

366.

(n2)−1(n2)−1

367.

(x4)−1(x4)−1

368.

(y−5)4(y−5)4

369.

(p−3)2(p−3)2

370.

(q−5)−2(q−5)−2

371.

(m−2)−3(m−2)−3

372.

(4y−3)2(4y−3)2

373.

(3q−5)2(3q−5)2

374.

(10p−2)−5(10p−2)−5

375.

(2n−3)−6(2n−3)−6

376.

u9u−2u9u−2

377.

b5b−3b5b−3

378.

x−6x4x−6x4

379.

m5m−2m5m−2

380.

q3q12q3q12

381.

r6r9r6r9

382.

n−4n−10n−4n−10

383.

p−3p−6p−3p−6

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

384.

45,000

385.

280,000

386.

8,750,000

387.

1,290,000

388.

0.036

389.

0.041

390.

0.00000924

391.

0.0000103

392.

The population of the United States on July 4, 2010 was almost 310,000,000.310,000,000.

393.

The population of the world on July 4, 2010 was more than 6,850,000,000.6,850,000,000.

394.

The average width of a human hair is 0.00180.0018 centimeters.

395.

The probability of winning the 20102010 Megamillions lottery is about 0.0000000057.0.0000000057.

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

396.

4.1×1024.1×102

397.

8.3×1028.3×102

398.

5.5×1085.5×108

399.

1.6×10101.6×1010

400.

3.5×10−23.5×10−2

401.

2.8×10−22.8×10−2

402.

1.93×10−51.93×10−5

403.

6.15×10−86.15×10−8

404.

In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was 2×104.2×104.

405.

At the start of 2012, the US federal budget had a deficit of more than $1.5×1013.$1.5×1013.

406.

The concentration of carbon dioxide in the atmosphere is 3.9×10−4.3.9×10−4.

407.

The width of a proton is 1×10−51×10−5 of the width of an atom.

Multiply and Divide Using Scientific Notation

In the following exercises, multiply or divide and write your answer in decimal form.

408.

(2×105)(2×10−9)(2×105)(2×10−9)

409.

(3×102)(1×10−5)(3×102)(1×10−5)

410.

(1.6×10−2)(5.2×10−6)(1.6×10−2)(5.2×10−6)

411.

(2.1×10−4)(3.5×10−2)(2.1×10−4)(3.5×10−2)

412.

6×1043×10−26×1043×10−2

413.

8×1064×10−18×1064×10−1

414.

7×10−21×10−87×10−21×10−8

415.

5×10−31×10−105×10−31×10−10

Everyday Math

416.

Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by 1.51.5 trillion calories by the end of 2015.

1. ⓐ Write 1.51.5 trillion in decimal notation.
2. ⓑ Write 1.51.5 trillion in scientific notation.

417.

Length of a year The difference between the calendar year and the astronomical year is 0.0001250.000125 day.

1. ⓐ Write this number in scientific notation.
2. ⓑ 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 11 by 2,598,9602,598,960 and saw the answer 3.848×10−7.3.848×10−7. 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 66 of her 5050 favorite photographs, she multiplied 50⋅49⋅48⋅47⋅46⋅45.50·49·48·47·46·45. Her calculator gave the answer 1.1441304×1010.1.1441304×1010. Write the number in decimal notation.

Writing Exercises

420.

1. ⓐ Explain the meaning of the exponent in the expression 23.23.
2. ⓑ Explain the meaning of the exponent in the expression 2−32−3

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.

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