Study Guide for Unit 2 Test

Roots, Powers, and Scientific Notation (by Yash Sony)

Square roots, Cubed roots, Perfect squares, Perfect cubes, and Non-Perfect squares

Square Rooting is just finding out what times itself equals what is in the radical ( square root box)

For example: √81=9 because 9 times 9=81


Square Rooting to the third power is just finding out what times itself 3 times is the number the square root box

For example: ∛27=3 because 3 times 3=9 and 9 times 3=27


Perfect Squares:

Perfect squares are numbers whose roots are rational numbers because a square has the same length sides all around.To get the square root of a number, you must find out what number times itself equals the number in the radical.

√1=1

√4=2

√9=3

√16=4

√25=5

(and so on)


Real Life Perfect Square Problem:

Lets say your dad wanted to build a birdhouse and needed to cut 2 pieces of wood for 2 sides of a bird house with an area of 144 inches per piece of wood. he needs to know how long to cut the sides, so you would need to find the square root of 144. Remember that a square root is a number which you have to multiply 2 of the same numbers to get. Ok, lets start at 10. 10 times 10=100, so that's not the answer. Lets try 11. 11 times 11 is 121, so that's not the answer either. 12 must be the answer. 12 times 12 is 144, so your dad has to cut the 2 pieces of wood 12 inches by 12 inches.


Perfect Cubes:

Perfect Cubes are numbers whose roots to the 3rd power are rational numbers because a cube has the same length sides all over. To get the cubed root of a number, you must find out what number times itself 3 times equals the number in the cubed radical.

∛1=1

∛1=1

∛8=2

∛27=3

∛64=4

∛125=5

(and so on)


Real Life Perfect Cube Problem:

Lets say you had a math project to make a perfect cube out of paper. Your teacher says the volume of the cube must be around 64 inches. You don't know how to measure the volume of a paper cube, so you must find the cubed root of 64. Lets start with 2. 2 times 2=4. 4 times 2=8, so that isn't the answer. Lets move onto to 3. 3 times 3=9. 9 times 3=27, so that isn't the answer either. Lets try 4 then. 4 times 4=16. 16 times 4 is 64, BINGO! So you must make each of the six sides of paper around 4 inches by 4 inches to get a paper cube with a volume of about 64.


Non-Perfect squares:

A non-perfect square is a square whose root is not a whole number.

√10=3.16

√11=3.32

√12=3.46

√13=3.61

√14=3.74

√15=3.87

(and so on)

To estimate a non-perfect square root, find the square roots of the numbers just below and above the non-perfect square. Lets pick √55. √49 and √64 are just above and below √55, and the square root of the two are 7 and 8. To find the √55, you have to keep on trying to pick in between 7 and 8. Lets pick 7.5. 7.5 times itself is 56.25, which is close to √55. Lets try a little bit lower with 7.4. 7.4 times itself is 54.76, so the answer is between 7.5 and 7.4. 7.4 is probably the closest we can get without going into too much detail. So lets say √55=about 7.4.


If you want to find a non-perfect square, start at √1. We know that √1=1, so we go to 4. Why 4? Well because we know √4=2. So √2 and √3 are non-perfect square. This pattern goes on and on adding 2 more non-perfect square each time you find a perfect square.

For Example:


√1=1
√2=1.4142135623731

√3=1.73205080756888

√4=2

√5=2.23606797749979

√6=2.44948974278318

√7=2.64575131106459

√8=2.82842712474619

√9=3


As you can see, there are two more non-perfect squares after √4 than √1. The list goes on like this forever.

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Exponent Powers

There are 7 exponent powers: product of powers, power of powers, power of a product, quotient of powers, power of a quotient, zero exponents, and negative exponents


Product of Powers:

Keep the base but add exponents.

B^N times B^M = B^(N+M) 4^2 times 4^2 = 4^(2+2)


Power of a Power:

Keep the base but multiply the exponents.

(B^N)^M = B^(N times M) (4^2)^2 = 4^(2 times 2)


Power of a Product:

When a product is raised to a power, each piece is raised to power.

(B times N)^M = B^M times N^M (4 times 2)^5 = 4^5 times 2^5


Quotient of Powers:

Keep base but subtract the exponents.

B^N/B^M = B^(N-M) 4^5/4^2 = 4^(5-2)


Power of a Quotient:

Multiply outer exponent by each base.

(B/N)^M = B^M/N^M (4/2)^5 = 4^5/2^5


Zero Exponent:

When anything, except 0, is raised to the zero power, it is 1.

B^0 = 1 (if B does not equal 0) 100^0=1 (0^0=0)


Negative Exponent:

Change the number into a fraction and make it positive.

B^-N (negative N) = 1/B^M (positive M) 4^-2 = 1/4^2


Real Questions:

1. 4^2 times 4^2 = 4^(2+2) = 4^4 = 256

2. (3^2)^2 = 3^(2 times 2) = 3^4 = 81

3. (5 times 2)^2 = 5^2 times 2^2= 25 times 4 = 100

4. 4^4/4^2 = 4^(4-2) = 4^2 = 16

5. (2/1)^2 = 2^2/1^2 = 4/1 = 4

6. 145,340,601,756^0 = 1

7. 3^-2 = 1/3^2 = 1/9


Real Life Exponent Powers Problem:

If you wanted help your mom repair the fence, she could tell you to get 12 pieces of 2^2 foot by 2^2 foot long wood with your dad from Home Depot. First, you would need to know how long that is before buying it. So 2^2 times 2^2 = 2^(2+2) = 2^4 which, in conclusion = 8. You would need to get 12 pieces of 8 footy long pieces of wood from Home Depot with your dad.

Scientific Notation

Scientific Notation:

Scientific notation is writing really big or really small numbers in a simpler form, and there are rules with scientific notation.

Rules:

1.When converting from scientific notation to standard notation by a positive exponent, you must make the number bigger.

2.When converting from scientific notation to standard notation by a negative exponent, you must make the number smaller.

3.When converting from standard notation to scientific notation, you must make the number that is being multiplied by 10 be between 1 and 10.

4.When subtracting or adding, you must make 1 exponent match the other by making it bigger or smaller, also changing the number that is being multiplied by ten. For example, if you need to make 2 times 10^3 match 2 times 10^4, then you can make 10^4 smaller, or 10^3 bigger. If you make !0^3 bigger, then 2 must get smaller, so 2 changes to 0.2 and 10^3 changes to 10^4. If your trying to make 10^4 smaller, 2 must get bigger, so 2 becomes 20, and 10^4 becomes 10^3.

5.The powers of exponents still apply to the exponents.


Instead of writing 1,000,000, you could just write 1 times 10^6

All your doing is adding zeroes, like 2 times 10^2 = 200 and 2 times 10^-2 = 0.002

All your doing is adding zeros based on the exponent above the 10

So 10^4= 100,000 as you can see, it just added four more zeros 10(0,000).

It's the other way around for negative exponents

For Example:

2 times 10^-4 = 0.0002

you add zeroes the other way, make the exponent negative, and make it a really small number.It also added three zeros this time because it counted the 2 as a zero.


If you want to change a number into scientific notation, count the numbers to the left or right until the number that is being multiplied by 10 is between 1 and 10

For example:

567 = 5.67 times 10^2 or 0.0567 = 5.67 times 10^-2


Multiplying In Scientific Notation:

(2 times 10^2) times ( 2 times 10^2) = 40,000 How? Lets take it step by step

First, place each number in it's group (2 times 2) times (10^2 times 10^2) The Power of a Power rule still applies to the both of the 10's

Second, multiply in the parenthesis (2 times 2 = 4) times (10^2 times 10^2 = 10^4)

Third, multiply using scientific notation 4 times 10^4= 40,000 just adding 4 zeroes


If your multiplying exponents with negative numbers, the only thing that changes is the outcome gets smaller instead of bigger, and you just count until the number you are multiplying by 10 is in between 1 and 10. Lets take it step by step

( 4 times 10^-3) times (7 times 10^-2)

Place each number in its group (4 times 7) times (10^-3 times 10^-2)

Multiply in the parenthesis (4 times 7 = 28) times (10^-3 times 10^-2 = 10^-5)

Multiply using scientific notation, but the number that is being multiplied by 10 is not in between 1 and 10, so we need to make it smaller, but if we make it smaller, we must make the exponent bigger, but is negative, so, for example, if you add-5+1, the outcome is -4. So our scientific notation is 2.8 times 10^-4 = 0.00028


Division In Scientific Notation:

(10 times 10^4) / (5 times 10^3) = 20 How? Lets take i t step by step.

Like multiplying, put the numbers in their groups (10 / 5) times (10^4 / 10^3) remember the Quotient of Powers rule still apply.

Then you DIVIDE the parenthesis (10 / 5 = 2) times ( 10^4 / 10^3 = 10^1)

Last, you multiply using scientific notation 2 times 10^1 = 20


If you divide using negative exponents, like multiplying, the only thing that changes is the outcome getting smaller instead of bigger. Lets use this problem:

(5 times 10^-5) / ( 2 times 10^2)

First, put the numbers into their groups (5 / 2) times (10^-5 / 10^2)

Second, you divide the parenthesis, but remember the Quotient of Powers rule still apply.(5 / 2 = 2.5) times (10^-5 / 10^2 = 10^-7) wait, why does 10^-5 / 10^2 = 10^-7? well because of the Quotient of Powers rule. When dividing exponents, you keep the base and SUBTRACT the exponents, but since one exponent is negative, subtracting a negative from a positive is like adding a negative and a negative. So, moving on, third, you multiply using scientific notation 2.5 times 10^-7 = 0.00000025


Addition In Scientific Notation:

(2 times 10^4) + (5 times 10^5) = 520,000 How? Lets take it step by step

First, you have to make the 10's have equal Exponents by making them bigger or smaller. The easier way is making 1 exponent bigger to equal the other, but in order to make the exponent bigger, the number that is multiplied by 10 must get smaller, so the 2 changes to a 0.2 and the 10^4 changes to a 10^5.

Second, you have to put the numbers into their groups, so (0.2 + 5) times 10^5

Third, you ADD (0.2 + 5 = 5.2) times 10^5

Fourth, you multiply using scientific notation 5.2 times 10^5 = 520,000


When adding using negative exponents, the outcome just gets smaller instead of bigger.Lets get an example.

(4 times 10^-2) + ( 3 times 10^-3) remember to put the numbers in their groups and make the exponents equal, this time, make the exponent SMALLER because they are negative, so if you make that side smaller, you have to make the number the number that is multiplied by 10 bigger (40 + 3) times 10^-3 now you add and multiply using scientific notation, but the number is not between 1 and 10, so you have to make it smaller by making the exponent bigger, but since the exponent is negative, instead of becoming 10^-4, it becomes 10^-2 again 4.3 times 10^-2 = 0.043


Subtraction In Scientific Notation:

(5 times 10^4) - (3 times 10^3) =47,000 How? Lets take it step by step

First you make the exponents equal, so lets make 10^4 smaller, but the number that is multiplying by 10 must get bigger (50 times 10^3) - (3 times 10^3)

Second, you put the numbers in their groups (50 - 3) times 10^3

Third, you SUBTRACT the parenthesis (50 - 3 = 47) times 10^3

Fourth, make sure the number isn't too big or small and change the number the is being multiplied by 10 and the exponent 4.7 times 10^4

Fifth, multiply using scientific notation 4.7 times 10^4 = 47,000


When subtracting using negative exponents, like adding, the only thing that changes is that the outcome will become smaller instead of bigger. Lets use this

(8 times 10^-2) - (6 times 10^-3) group the number and remember that you have to make the exponents match, but these are negative, so do the opposite

(80 - 6) times 10^-3 subtract the parenthesis and make the number that is being multiplied by 10 smaller and the exponent bigger, but since the exponent is negative, it will be 10^-2 instead of 10^-4 (80 - 6 = 74) times 10^-3 7.4 times 10^-2

finally, multiply using scientific notation 7.4 times 10^-2 = 0.074


Real Life Scientific Notation Problem:

If you needed to know how many pages were in all of the Harry Potter books, but the number was too large, you could write it in scientific notation. Lets say that The Harry Potter series has 2,037 pages. Write that number in Scientific Notation.


2.037 times times 10^3