©Judo Math Inc.

EXPONENTS

©Judo Math Inc.

8th Grade

Orange Belt Training – Problem Solving Discipline

Do You Have the Powerx to Get √𝑹𝒂𝒅𝒊𝒄𝒂𝒍?

This Belt addresses a very important set of operations used in algebra: Power & Roots. Power & Roots are another set

of inverse operations used in mathematics and just like + & - and * & ÷ are inverse (opposite operations) so too are

Powers & Roots. You may be wondering what are powers and roots…well, it is something you have certainly seen

before; most likely, in terms of Squares & Square Roots…..like 52 read five squared equals 25 and its square root

√25 = 5 You see 5 is the base and the square root of 25 is 5…..the square, that is the 2 is the Power and the √ is

the square Root.

Let’s Break it down:

𝑥𝑦 = √

You will see all sorts of numbers and variables with an exponent (or power) and then you’ll be asked to find their roots,

numbers Roots, specifically its Square Root (√ ) and its Cube Root (√3

). For example,

Exponent Form What it Means (Expanded Form) Square Root Equals

32 3 ∙ 3 = 9 √9 3

56 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 = 15,625 √15,625 125 or 53

An Exponent or Power tells you “How many times” to multiply the base by itself. A Radical (square or cube root) asks what number was squared or cubed to get radicand. Confusing? Yes, we know. But after you finish this belt…You will be a true master! Good luck grasshopper.

Order of Mastery - Exponents (8.EE.1-4)

1. Foundations of Integer Exponents & Powers 2. Working & Simplifying Exponential Expressions 3. Expressing numbers in Scientific Notation 4. Operations with Scientific Notation 5. Simplifying Radicals- Square & Cube Roots

Exponent or Power Radical Sign

Radicand Base

Standards Included:

8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For

example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p,

where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small

perfect cubes. Know that √2 is irrational.

8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or

very small quantities, and to express how many times as much one is than the other.

8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal

and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of

very large or very small quantities

o

1

1. Foundations of Exponents

Discovering Exponents: It’s POWERful

1. Complete the table by filling in missing information.

Problem Expanded Form Simplified Answer

3 x 3 x 3 x 3 x 3

73

𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦

2?𝑤? 32w6

See a Pattern? What’s the Definition of an Exponent (in your best words).

2. Complete the table by filling in missing information.

Problem Expanded Form Simplified Answer

(5 ∙ 5 ∙ 5 ∙ 5) ∙ 5 ∙ 5 ∙ 5 57 = 78,125

25 ∙ 24

45𝑤4 = ? 𝑤4

43𝑤 ∙ 42𝑤4

See a Pattern? What’s the Rule for Multiplying Same Base Exponents (in your best words).

2

3. Complete the table by filling in the missing information.

Problem Expanded Form Simplified Answer

98

93⁄ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9

9 ∙ 9 ∙ 9 ∙ 9 ∙ 9

3 ∙ 3 ∙ 3 ∙ 3

3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3

1

33 =1

27 or 3-3

𝑔9𝑥3

𝑔7𝑥7

65

65

See a Pattern? What’s the Rule for Dividing Same Base Exponent(in your best words)?

See something peculiar above? What does it mean to have a negative exponent? Give an example and show expansion.

4. Complete the table by filling in the missing information.

Problem Expanded Form Simplified Answer

(10𝑦3)4 (10𝑦3)(10𝑦3)(10𝑦3)(10𝑦3)

(52𝑥4)3

(3𝑥2𝑦)(3𝑥2𝑦)(3𝑥2𝑦)

See a Pattern? What’s the Rule for Raising to a Power (in your Best Words)?

A base to with a zero exponent or power is always, always, always what? Prove with an example?

3

The Rules of Working with Exponents (makes simplifying easier)

Essential Requirement to using these rules is that you must have the _________ Base!

Rule #1 Product Rule Example

(xm)(xn) = xm + n (52)(55) = 52 + 5 = 57

Practice: 43 · 45 = 7 · 74 = m2 · m7 =

_______________________________________________________________________

Rule #2 Division Rule Example

xm ÷ xn = xm – n x7 ÷ x2 = x7 – 2 = x5

Practice: 49 ÷ 45 = 83 ÷ 8 = b7 ÷ b4 =

________________________________________________________

Rule #2a Negative Exponent Rule Example

x –m = 5 -3 = =

Practice: 7-5 = w-11 =

_______________________________________________________________________

Rule #3 Raising to a Power Rule Example

(xy)m = xmym (4x)3 = 43 · x3 = 64x3

Practice: (5x4)3 = (24w3y)5 =

_________________________________________________________________

Rule #4 Power of Zero Rule Example

𝒙𝟎 = 𝟏 45,789,3240= 1

mx

135

1

125

1

4

2. Simplifying Exponential Expressions….Fancy Stuff!

Now, let’s try your application of the the exponent rules….in simplifying.

1. 42

2. 20

3. 53

4. 63

5. 62

6. 30

7. 6-5

8. 54

9. (-5)0

10. 44

11. (-12)-3 =

12. 72

13. Rewrite the following in its equivalent expression:

a) 7-3 = b) 8-2x-5 = c) 1

53𝑦5 = d) 43𝑤−7 =

14. 56 ∙ 512 = 15. 72 ∙ 76 ∙ 73 = 16. 4−2 ∙ 4−6 =

17. 210 ∙ 28 = 18. 33 ∙ 3−9 ∙ 32 ∙ 38 = 19. 8−3(84)(8−5) =

20. 9−9 ∙ 90 ∙ 94 ∙ 9−2= 21. 102 ∙ 153 ∙ 104 = 22. 𝑥6 ∙ 𝑥−4 ∙ 𝑥8 =

23. 5𝑥3(10𝑥5)(3𝑥4) = 24. 72𝑤4𝑦3(52𝑤−3𝑦) = 25. 83𝑘−6𝑚10(8−5𝑘−2𝑚−6)

Power of a Power (it looks worse than it really is) Distribute the Power….

26. (121𝑥4𝑦3)3 = 27. (34𝑤5𝑧−2)5 = 28. (5−2𝑦3𝑤−4)−5 =

5

29. (-3q-6)3(-q6x-6)2 30. (-d6)3(-2d4n-4)2 31. (-4q-5)3(-q4s-4)5

32. (-5u3j-3)4(-u4j4)5 33. (-r-5t5)5(-r-6t-6)3(r-2)5

Division makes it even more fun!! Woo-hoo!!

34. 158

155= 35.

714

711= 36.

512𝑥4

514𝑥2=

37. = 38. 125𝑘9𝑚4

52𝑘6𝑚6= 39.

−𝑑3

−10𝑑6=

40. 8−10

8−9 ÷ 89

85 = 41. 7−5 ÷ 7−3 = 42. 52𝑥4

103𝑥−5÷

𝑥7

103𝑥−3=

Applying a Power to a Fraction….distribute to numerator and denominator!

43. (53

45)3

= 44. (𝑥𝑦3

𝑥5𝑦2)5

= 45. (3𝑤−3

35𝑤7)4

=

9xy-4

3x-2y

6

The Power of 10!

The Number 10 is a powerful number in mathematics especially when you take it a certain Power. The 10’s

units is seen in the Place Value System, used with Percents (x100), the metric system, logs in Algebra, and of

course Scientific Notation. Let’s look at 10 to various powers…

Positive Power Negative Power

101 = 10 10-1 = .1

102 = 100 10-2 = .01

103 = 1000 10-3 = .001

104 = 10,000 10-4 = .0001

Now you fill in the rest….

105 = ___________ 10-5 = ____________

106 = ____________ 10-6 = ____________

See a Pattern? What effect does the Power on the 10 do to the answer? Explain

Once you know, it is easy to do the following problems.

1. 4.92 x 104 = 2. 3.87 x 106 = 3. -7.89 x 107 = 4. 3.94 x 10-4 =

5. 1.45 x 10-6 = 6. 8.2 x 1012 = 7. -2.38 x 10-8= 8. 9.1 x 10-11 =

9. So, what does the Power on the 10 tell you about the answer? How do you get your answer?

10. Why is 3.87 x 10-5 and 3.87 ÷ 105 the same answer? Explain.

7

3. Scientific Notation

An Application in Exponents

Named because it is used mostly in “Scientific” applications and “Notation” because it is shorthand way to write very

large numbers.

How would you like to write or work with this number 734,824,005,000,912….that is a crazy large Number….scientific

notation makes it easier!

Scientific Notation is a convenient way to mathematically abbreviate very large and very small numbers. It is basically

taking a number in its units digit form and multiplying it by a power or negative power of ten; that is, counting the

number of 10’s moving the decimal point between the first and second number.

For example: 50,000,000 can be obtained by multiplying 5 by 10, and again by 10, and again by 10, and so on

until 10 has been used as a multiplier seven times. The scientific notation form of this number is to write 5 x 107

The same can be done with very small numbers but by dividing by 10 (that is, negative power of 10).

For example, .0000000567 can be obtained by multiplying 5.67 by 10-8 (or by dividing 5.67 by 108). The

scientific notation form of this number is to write 5 x 10-4 Some other examples

425,000,000,000 would equal 4.25 x 1011

.00000037 would equal 3.7 x 10-7

Now You Try some…..

Convert to Scientific Notation:

8

1. 34,000,000 2. 2,000,000,000 3. .000006

4. .00000078 5. 785,000,000 6. -.000038

7. 897,300,000,000,000 8. .00000000768 9. -567,000,000,000,000

Convert to the actual number:

10. 2.3 x 104 11. 1.3 x 107 12. 56 x 10-5

13. 4.5 x 10-8 14. 4.79 x 107 15. -3.2 x 10-6

Multiplying Numbers in Scientific Notation….use your Exponent Rules (multiply the numbers and ADD your

exponents for base 10)

16. (2 x 108) (4 x 105) = 17. (3 x 1015) (2.8 x 107) = 18. (7 x 108) (9 x 105) =

19. (-5.6 x 1010) (3.23 x 106) = 20. (7.8 x 109) (-3.4 x 105) =

Dividing Numbers in Scientific Notation….use your Exponent Rules (divide the numbers and SUBTRACT your

exponents for base 10)

21. (1.2 x 109) ÷ (4 x 107) = 22. (8.8 x 1015) ÷ (6.2 x 108) =

23. (3.8 x 1012) ÷ (9 x 1015) = 24. 8.2 𝑥 105

2.8 𝑥 109=

9

4. Solving Problems in Scientific Notation

Over the course of your work as a student scientist, you will certainly encounter VERY LARGE numbers and

VERY SMALL numbers. Instead of writing every digit of the VERY LARGE or VERY LARGE number…..use

estimates in scientific notation by using the first one or two digits.

For example, you see a number like 956,783,556,000,034. To work with this, estimate the number in scientific

notation: 9.6 x 1014

You have rounded the number but it works just the same!

Now you try some:

1. Which is greater 6 x 108 or 5 x 107? How do you know?

2. Which is greater 2 x 10-5 or 8 x 10-6 ? How do you know?

3. Write in order from smallest to largest: 6.45 x 108 , 2.3 x 1012 , 4.5 x 10-4, 3.8 x 10-6

4. What is the missing exponent in the following problems:

a) 103 𝑥 10? = 108 b) 104 𝑥 10? = 10−6 c) 10−8 𝑥 10? = 10−6

5. Would you rather have 2 x 1028 dollars or 2 x 10-28 dollars? What is the difference?

6. What is the ratio of Milky Way radius to our solar system radius given that, the distance from pluto to sun is

5.9×10 12 meters and the Milky Way disk radius is. 3.9×1020 meters. Round the coefficient to the nearest tenth.

10

7. The mass of an electron is about 1/8th the size of the mass of a neutron. If the mass of a neutron is 1.67 x 10-27 kg.

What is the mass of an electron? Express in scientific notation. How did you get your answer.

8. 2,800,000,000 cubic centimeters of water flow over Niagara Falls in just one minute. How much water flows over the

Falls in a year? (remember use estimates and what other information do you need)

9. The speed of light 3 x 108 meters per second. If the sun is 1.5 x 1011 meters from earth, how many seconds does it

take light to reach the earth.

10. Assume that there are 20,000 runners in the New York City Marathon.. Each runner runs a distance of 26 miles. If

you add together the total number of miles for all runners, how many times around the globe would the marathon

runners have gone? Consider the circumference of the earth to be 2.5 x 104 miles.

11. The size of a period is about .0006 meters in length. The size of a bacteria cell is approximately .0000021 meters in

length. About how many times greater is the period?

11

Slightly Scarier Exponents

Know your Exponent Rules….check out these additional rules (based on the other rules).

= (xy)m = xmym

( )-2 = ( )2 x –m = x0 = 1

_________________________________________

Time to step it up a notch…

1. (-5d-6)(-6d-3s3) 2. 4v3(z3v3) 3. (4i-2)(-4i3x-3)(-10i-2)

4. (-u-2x-2)(-11u4x4)(-u-2) 5. (-8s-6)(8s-6q-6) 6. (-9n-2)(-7n3r3)(-8n-4r-4)

7. (-3q-6)3(-q6x-6)2 8. (-d6)3(-2d4n-4)2 9. (-4q-5)3(-q4s-4)5

10. 11. 9𝑗6𝑘3

10𝑗5𝑘 12.

n

m

x

x

m

n

x

x

n

m

x

xm

n

x

xmx

1

5xy

z-2

æ

èç

ö

ø÷

3 3

24

3

8

7

yg

g

12

13. −11𝑢5𝑔2

−9𝑢3𝑔4 14. (-20x2)0 ÷ (8x2)2 15. (16x) ÷ (-32x)3

16. (2x4)3 ÷ (-2x4)4 17. 18. (7x2)0 (-31x4) (31x4)3

19. (-2x)0 (24x)4 20. (-3x3)2 (-32x4)3 21. (-18x3)4 (23x2)0

Challenger….these are just plain torture!

22. 23. (−3𝑑3𝑦4)(9𝑑4)

(−8𝑑5𝑦3)(−2𝑑3𝑦4)

24. 25.

(2x4 )(6xy3)

4y3

(4x-1y-2 )-3(xy3)2

(3x-1y3)2

15m3n-2p-1

25m-2n-4

æ

èç

ö

ø÷

-3 0

63

542

))((7

3)(4()9(÷÷ø

öççè

æ

--

---

--

xyyxxy

yxxyx

13

5. Time to get √𝑹𝒂𝒅𝒊𝒄𝒂𝒍

Whew! Aren’t you glad the exponents are over? Well, now it is time to go in reverse-

Simplifying Radicals….specifically, we’ll be working with Square Roots and Cube Roots

Square Root (√ )- of a number is value, that when multiplied by itself, gives the number. For example, 4 x 4

= 16, so the square root of √16 is 4. 4 is called the root.

- Basically, it is asking what number squared equals the number under the radical?

You gotta know your perfect squares: 4, 9, 16, 25, 36,….etc. These are nice, pretty whole numbers when

you take their square root.

Cube Root (√𝟑

)- of a number is a special value, that when used in multiplication three times, gives that number.

Example, 3 x 3 x 3 = 27 so the cube root √273

= 3

List out all the perfect squares from 1 to 15….i’ll get you started

√1 = 1 , √4 = 2, √9 = 3 , ….

List out all the perfect cube roots from 1 to 10….I’ll get you started

√13

= 1, √83

= 2 , √273

= 3 ,

Radicands, Radicals and everything else you never wanted to know about Square Roots

We’ve dealt with squaring a number…well the square root is just the opposite. So if 3 squared equals 9,

written as 32 = 9. Then that means that 3 is a square root of 9.

The radical symbol is that funky looking thing that tells you to take the square of a number. It looks like

this:

√

What’s inside that funky thing is called the radicand. That could be a number, expression, or a variable.

We’ll deal with the numbers for now.

In this example, 9 is the radicand: 9 = 3 and you would read it as: the positive square root of 9 is

3

Here’s the tricky part. All positive real numbers have TWO square roots. The positive square root and

the negative square root . So in the above example, -3 is also a square root of 9, since (-3)2 = 9

Positive square root of 25 is 5 25 = 5

Negative square root of 25 is -5 - 25 = -5

So you may see it written like this on some standardized test: 25 = 5

Some other useful things to know about square roots:

-A perfect square is a whole number that is the square of an integer. So the square root of a

perfect square is always a nice full number (integer, which means no fractions or decimals) 49, 25, 4, 16

are all examples of perfect squares

-If you get a problem like this one: 3 = ? The answer will be an irrational number since 3 is

not a perfect square. (the answer is 1.73… by the way)

-If you come across a problem where the negative sign is INSIDE the radical, that means there’s

no real solution, because the square of a number is always positive. 25 has no real solution.

2

Now You Try Some:

1. Circle the following numbers that are perfect squares:

10 100 49 64 3 4 27 24 16 ½ 9/4 111 225 120 81

2. Find the square roots of the following (remember answer always with ± )

√144 = √36 = √256 = √169 = √625 = √16

64 =

Evaluate the following expressions…without using a calculator (remember the Yellow Belt?).

Approximate to the nearest tenth.

3. √5 = 4. √10 = 5. √7 =

6. - √20 = 7. √15 = 8. √70 =

Evaluate the expression: √𝑏2 − 4𝑎𝑐 for the given values

9. a = 4, b = 5, c = 1 10. a = 2, b= 4, c = -6

11. a = -2, b= 8, c= -8 12. a= -5, b= 5, c= 10

Evaluate the radical expression when a = 2 and b = 4

13. √𝑏2 + 10𝑎 = 14. √𝑏2−8𝑎

4 =

15. √𝑏2+42𝑎

𝑎 = 16.

36− √8𝑎

𝑏 =

3

When Things Aren’t Exactly PERFECT…

Wouldn’t it be nice if all number were PERFECT SQUARES (or cubes), but that only exists in math fantasy

land. Numbers like √2, √35, √125 and many others are NOT Perfect (called Imperfect Squares). In

Fact, you learned earlier that these numbers are part of a group of numbers called Irrational Numbers.

So, how do we handle these numbers??

Now that we know what that funky symbol means, it’s time to step it up a notch. When we’re

dealing with these radicals, we need to remember to always simplify, just like with fractions (ahhh!!!!

fractions!!!)

So here’s the deal. The simplest form of a radical:

-has no perfect square factors other than 1 in the radicand

-has no fractions in the radicand

-has no radicals in the denominator of a fraction

First you need to understand the Product Property of Radicals, which is defined as follows:

ab = a b

Break up the number into possible Factors of that number and look for the Perfect Squares:

Here are some examples of that property (the key is to look for the Perfect Squares):

√20 = 4 ∙ 5 which can be simplified as 2 5

x16 = 16 ∙ x which can be simplified as 𝟒√𝒙

98 = 2 ∙ 49 which can be simplified as 7 2

√𝟕𝟐 = √𝟖 ∙ √𝟗 = √𝟒 ∙ √𝟐 ∙ √𝟗 = which can be simplified as 2 ∙ 3 ∙ √2 = 6√2

Keep breaking it down till you find Perfect Squares then take the Square Root of those Numbers, the

other numbers stay under the √

Now you Try some…..

4

Your turn!!! SIMPLIFY the following expressions:

1. 44 2. 54 3. 18

4. 56 5. 27 6. 27

7. 63 8. 200 9. 90

10. √𝟏𝟐𝟓 11. 𝟐√𝟐𝟒 12. 𝟓√𝟓𝟎

13. 𝟕√𝟖𝟎 14. 𝟏𝟎√𝟐𝟖 15. 𝟑√𝟒𝟖

Let’s Simplify some Variables with Radicals

13. √𝑥2 14. √𝑥4 15. √𝑥6 16. √𝑥8 17. √𝑥10

18. What Pattern do you notice for problems #13 – 17?

19. Simplify the following….what do you notice or what pattern emerges now?

√𝑥 √𝑥3 √𝑥5 √𝑥7 √𝑥9 √𝑥11

5

Now, you may see Radicals with Fractions. Yikes!!! Fractions!! Say it Ain’t So…yes, it’s called the

Quotient Property of Radicals

b

a =

b

a

Some examples of that include:

16

9 =

16

9 =

4

3

4

x =

4

x =

2

x

Here’s a tricky example. You can’t have a radical in the denominator, so to simplify, you must get rid of

it:

2

1 =

2

1 =

2

1 But that’s not our answer….since we can’t have that radical on the

bottom.

To get rid of it, just multiply the entire expression by the value of that radical:

2

1 x

2

2 =

22

21 =

4

2 =

2

2

PUTTING IT ALL TOGETHER: Example: If you were asked to simplify the following expression:

50 You would first factor 50 = 105 = 25 x 2 = 5 2

Here, you try a few:

10. 16

4 11.

49

9 12.

25

4

13. 5

1 14.

6

5 15.

5

3

6

Dude, Let’s Get Totally √𝑹𝒂𝒅𝒊𝒄𝒂𝒍

Put it all together….

1.

 

24 2.

 

98 3.

 

32

4.

 

196 5.

 

9 50 6.

 

3 18x4

7.

 

4 72 8. √𝑥25 9.

 

32a9b3

10.

 

72c9d13 11.

 

49a4b3

9 12.

 

16d8x 4

121 f 7x 6

13.

 

(x - 7)2 14. 24xy2

15. b27

16. What is a convenient rule to help you find the roots of a radicand especially

variables raised to a power?

7

Just to Test How Radical you are at Solving Exponents & Radicals?

Remember, taking a square root undoes squaring and, Squaring undoes taking the

square root.

1. If x2 = 81 , what is the value of x? 2. If y2 = 125, what is the value of y?

3. Solve w2 = 400 4. What is the value of 25 = √𝑦 ?

5. Solve √𝑥 = 12 6. What is the value of x when x2 = 150

7. I am putting in square tiles in my bathroom. If each tile has an area of 144 cm2. What is the

measurement of each side of the tile? How many tiles will I need to cover a floor space of 1.8

m2?

8. You know the Pythagorean Theorem a2 + b2 = c2 which tells you the sides of a Right

triangle. If a (one of the sides) = 13 and b (one of the legs of the triangle) = 15, what is the

value of c (the Hypotenuse)?

a=13

b=15

C = ?

8

Answers- Orange Belt: Problem Solving

Discover Exponents- Discuss in class/See Teacher

Rules of Exponents:

1. 16 2. 1 3. 125 4. 216 5. 36 6. 1 7. 1/1776 8. 625 9. 1 10. 256 11. 1/1728 12. 49 13. a) 343 b)x-7 c) 5-3y-5 d) 64/w7 14. 518 15. 711 16. 4-8 17. 218 18. 34 19. 8-4 20. 9-7 21. 106*153= 3,375,000,000 22. X10 23. 150x12 24. 1225 25. 8-2k-8m4 26. 20,736x12y9 27. 320w25z-10 28. 510y-15w20 29. 27q-6x-12 30. 4d26n-8 31. -64q5s-20 32. 625u32j8 33. –r-53t7 34. 153 35. 73 36. 5-2x2 37. 3x-1y-3 38. 5k3m-2 39. 1/10d3 40. 8-3 or 1/83 41. 7-2 42. 25/x 43. 59/415 44. X-20y5 45. 3-16w40

Power of 10

1. 49,200 2. 3,870,000 3. -78,900,000 4. .000394 5. .00000145 6. 8,200,000,000,000 7. -.000000238 8. .000000000091 9. How many places you move the decimal point 10. Mult & div. are inverses of each other, same thing

Scientific Notation

1. 3.4 x 107 2. 2 x 109 3. 6 x 10-6 4. 7.8 x 10-7 5. 7.85 x 108 6. -3.8 x 10-5 7. 9.0 x 1014 8. 7.68 x 10-9 9. -5.67 x 1015 10. 23,000 11. 13,000,000 12. .000056 13. .000000045 14. 47,900,000 15. -.0000032 16. 8 x 1013 17. 8.4 x 1022 18. 6.3 x 1014 19. -1.8 x 1016 20. 2.65 x 1015 21. 30 22. 1.42 x 107 23. 4.2 x 10-4 24. 2.9 x 10-4

Solving Problems in Scientific Notation

1. 6 x 108 2. 2 x 10-5 3. 3.8x10-6, 4.5x10-4, 6.5x108,2.3x1012 4. a) 5 b) -10 c) 2 5. 2x1028 bigger number 6. 6.6 x 107 7. 2.08 x 10-28 8. 1.4 x 1015 9. 500 seconds or 8.3 min 10. 20.8 times 11. 3000 times greater

Slightly Scarier Exponents

1. 30d-9s3 2. 4v6z3 3. 160i3x-3 4. -11x2 5. -64s-12q-6 6. -504n-3r-1 7. -27q-6x-12 8. -4d26n-6 9. -64q5s-20

10. 125𝑥3𝑦3

𝑧−6 11. 9𝑗𝑘2

10 12.

−512𝑔21𝑦6

343 13.

−11𝑢2

−9𝑔2 14. 1

64𝑥4 15. 16

323𝑥2 16. 1

2𝑥4 17. 3x5 18. 314x16

19. 331,776x4 20. 294,912x18 21. 104,976x12

Radicals, Radicands….about Square Roots (assume all answer with ±)

1. 100, 49, 64, 4, 16, 9/4, 225, 81 2. ± 12, 6, 13, 16, 25, 4/8 or ½ 3. 2.2 4. 3.2 5. 2.7 6. -4.5 7. 3.9 8. 8.4 9. 3 10. 8 11. 0 12. 15 13. 6 14. 0 15. 5 16. 2 or 10

When Things aren’t Exactly Perfect

1. 2√11 2. 3√6 3. 3√2 4. 2√14 5. 3√3 6. 2√15 7. 3√7 8. 10√2 9. 3√10 10. 5√5

11. 4√6 12. 25√2 13. 28√5 14. 20√7 15. 12√3 16. X, x2, x3, x4, x5 17. See Teacher 18.

√𝑥, 𝑥√𝑥, 𝑥2√𝑥, 𝑥3√𝑥, 𝑥4√𝑥, 𝑥5√𝑥 19. 2/4 or ½ 20. 3/7 21. 2/5 22. √5

5 23.

√30

6 24.

√15

5

9