How many different numbers between $\dfrac{1}{1000}$ and $1000$ can be written either as a power of $2$ or as a power of $3$, where the exponent is an integer?

Answers

Answer 1

Answer:

31

Step-by-step explanation:

Answer 2

Answer:

31

Step-by-step explanation:

Solution:

First we count the powers of $2$. We can compute $2^9=512$ and $2^{10}=1024$, so the largest power of $2$ which is smaller than $1000$ is $2^9$. Remembering that $2^{-n}$ is defined as $\dfrac{1}{2^n}$, we can see that the smallest power of $2$ which is larger than $\dfrac{1}{1000}$ is $2^{-9}$. Thus, we have the following list of powers of $2$:

\[2^{-9}, 2^{-8}, 2^{-7}, 2^{-6}, 2^{-5}, 2^{-4}, 2^{-3}, 2^{-2}, 2^{-1}, 2^0, \]\[2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^8, 2^9.\]

There are $19$ numbers in this list ($9$ with negative exponents, $9$ with positive exponents, and one more -- $2^0$).

We can count powers of $3$ in a similar way. The largest power of $3$ smaller than $1000$ is $3^6 = 729$. The smallest power of $3$ larger than $\dfrac{1}{1000}$ is therefore $3^{-6}$, or $\dfrac{1}{729}$. So we have these powers of $3$:$$3^{-6}, 3^{-5}, 3^{-4}, 3^{-3}, 3^{-2}, 3^{-1}, 3^0, 3^1, 3^2, 3^3, 3^4, 3^5, 3^6.$$We don't want to count $3^0$ again, though, because $3^0$ is $1$, which we already counted as $2^0$! There are $12$ other numbers in our list ($6$ with negative exponents and $6$ with positive exponents).

We don't have to worry about any other duplication between the lists, since $\allowbreak 2^1,2^2,2^3,\allowbreak \ldots,2^8,2^9$ are all even and $\allowbreak 3^1,3^2,3^3,\allowbreak 3^4,3^5,3^6$ are all odd. So, we have $19+12 = \boxed{31}$ different numbers in our two lists.


Related Questions

Find the perimeter and area of the polygon with the given vertices P(-3,4), Q(1,4), R(-3,-2), S(3,-2)

Answers

Answer:

Step-by-step explanation:

Find the perimeter and area of the polygon with the given vertices P(-3,4), Q(1,4), R(-3,-2), S(3,-2)

We find the length of the sides of this polygon using the formula:

√(x2 - x1)² + (y2 - y1)2

Where (x1, y1) and (x2, y2)

P(-3,4), Q(1,4)

= √(1 -(-3))² + (4 - 4)²

= √4²

= √16

= 4 units

Q(1,4), R(-3,-2)

= √(-3 - 1)² + (-2 - 4)²

= √-4² + -6²

= √16 + 36

= √52

R(-3,-2), S(3,-2)

= √(3 - (-3))² + (-2 - (-2))²

= √6²+ 0

= √36

= 6 units

P(-3,4), S(3,-2)

= √(3 - (-3))²+ ( -2 - 4)²

= √6² + -6²

= √36 + 36

= √72 units

What are the dimensions of the simple shapes you obtained in part C? List dimensions for all your decompositions (sets of shapes).

Answers

Answer:

Option 1

Figure Length (feet) Width (feet)

small rectangle 14 6

large rectangle 20 7

Figure Base (feet) Height (feet)

triangle 6 6

Option 2

Figure Length (feet) Width (feet)

small rectangle 6 7

large rectangle 14 13

Figure Base (feet) Height (feet)

triangle 6 6

Step-by-step explanation:

Answer:

The volume of a rectangular prism whose dimensions are l, w, and h is

V = l × w × h.

Option 1:

The length (l), width (w), and height (h) of the first prism are  inches, 26 inches, and  inches, respectively, so the volume of the first prism (V1) is

Step-by-step explanation:

Spanky's Sporting Goods is having a sale. All merchandise is 20% off the original price. The original price on a tennis racket is $78.99.

What is the sale price?
A
x
$15.80
B
x
$63.19
C
x
$157.98
D
x
$58.99

Answers

Answer:

B. $63.19

Step-by-step explanation:

You will pay $63.19 for a item with original price of $78.99 when discounted 20%. In this example, if you buy an item at $78.99 with 20% discount, you will pay 78.99 - 15.798 = 63.19 dollars.

6) Triangle RST has vertices R(2,3), S(5. -1), and T(3, -4). The triangle is rotated 90° counterclockwise about the origin. Which are the coordinates of the image of vertex T?​

Answers

Answer:

T'(4, 3)

Step-by-step explanation:

The rule of 90° counterclockwise rotation about the origin:

(x, y) → (-y, x)

Apply the rule to the point T:

T(3, -4) → T'(4, 3)

Answer:

A(x,y)____rotation through 90°. >A'(-y,x)

Step-by-step explanation:

T(3,-4))____rotation through 90°. >T'(4,3)

R(2,3))____rotation through 90°. >R'(-3,2)

S(5,-1))____rotation through 90°. >S'(1,5)


T-shirt Sale
Any 3 T-shirts for $14.50
$3.99
$6.99
$5.99
A) Tom bought these three T-shirts at the sale price of $14.50. How much money did he save compared
to the original total price of the T-shirts?

Answers

Answer:

add 3.99 + 6.99 + 5.99 = 16.97 divide 14.50 by 16.97 you will get .854 multiply by 100 it is %85.4 that is the percentage he paid then he saves 100 - 85.4 = %14.6

Evaluate the expression n^2-16

Answers

Is there a picture to the problem?

12
A scientist measured the wavelength of an orange light wave as 0.000000615 meters.
A second scientist measured the wavelength of a green light wave as 5.6 x 10-7 meters.
How much longer, in meters, was the orange light wave than the green light wave?

Answers

Answer:

it 5.5⋅10−^8m

Step-by-step explanation:

Unless I'm missing something important here, you can find the difference between the two wavelengths by subtracting one from the other. Since you're interested in finding how much longer the wavelength associated with the orange light is, subtract the wavelength of the green light from the wavelength of the orange light. You know that the two measured wavelengths are 6.15 ⋅ 10 − 7 m → orange light 5.6 ⋅ 10 − 7 m → green light Therefore, the difference between the two wavelengths will be Δ wavelength = 6.15 ⋅ 10 − 7 m − 5.6 ⋅ 10 − 7 m = 5.5 ⋅ 10 − 8 m

I need help on this and the person who answer this correctly gets a BRANLIST(answer if you know)​

Answers

The answer that you’re looking for is B

Answer:

A)

Step-by-step explanation:

Linear functions are those whose graph is a straight line. A linear function has the following form. y = f(x) = a + bx. A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y

Sales tax in Idaho is 6%. If your grocery bill before sales tax was $50, how much will you have to pay in sales tax?

How do I solve the problem? ​

Answers

Answer:

$3.00

Step-by-step explanation:

6%=0.06----Convert the percentage into a decimal

0.06*$50=$3.00----Multiply the decimal by the total grocery bill and get $3.00!

Hope this helped

Will mark brain list

Answers

Answer:

36

Step-by-step explanation:

6+6=12 12+12=24 9+6=15 15+24=39

In a survey 80 students were asked to name their favorite subject.30 students sad. English was their favorite subject. What percent of the students surveyed said that the English was their favorite subject.∛

Answers

Answer:

37.5%

Step-by-step explanation:

30/80=37.5 percent

Answer:

37.5% of students say English is their favorite subject

Step-by-step explanation:

30/80=0.375----Divide the amount of students that like English(30) by the total amount of students who took the survey.(80)

0.375=37.5%----Turn your decimal answer to a percentage to get 37.5%!

SOMEONE HELP!!!!!!!!!!!!!!!

Answers

Answer:

Q= 5c

Step-by-step explanation:

Q=5c ........................

In a lab experiment, 490 bacteria are placed in a petri dish. The conditions are such
that the number of bacteria is able to double every 29 hours. How many bacteria
would there be after 8 hours, to the nearest whole number?

Answers

Answer:

593

Step-by-step explanation:

[tex]490(2)^8^/^2^9 = 593.25[/tex]

593.25 ≈ 593

The number of bacteria after 8 hours is 593.

What is a word problem?

A word problem is a verbal description of a problem situation. It consists of few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

For the given situation,

Number of bacteria placed in petri dish = 490

Time required for bacteria to double = 29 hours

Number of bacteria after 8 hours is

⇒ [tex]490(2)^{\frac{8}{29} }[/tex]

⇒ [tex]490(2)^{0.2758}[/tex]

⇒ [tex]490(1.2107)[/tex]

⇒ [tex]593.25149[/tex]

⇒ [tex]593[/tex]

Hence  we can conclude that the number of bacteria after 8 hours is 593.

Learn more about word problems here

brainly.com/question/20594903

#SPJ2

Mikeply o binomial by a trinomial
(x-1) (x²+x+1)​

Answers

Answer:

See if this is helpful

Answer:

x³ - 1

Step-by-step explanation:

Given

(x - 1)(x² + x + 1)

Each term in the second factor is multiplied by each term in the first factor, that is

x(x² + x + 1) - 1(x² + x + 1) ← distribute parenthesis

x³ + x² + x - x² - x - 1 ← collect like terms

= x³ - 1

help me pls !!!!!!!!!!

Answers

Answer: What do u need help with?

Step-by-step explanation:

which number is correct

Answers

Answer:

5

Step-by-step explanation:

Side QR is equal to PS.

QR = 2x + 3

PS = 4x - 7

Since both sides are equal, here is the equation:

2x + 3 = 4x - 7

x = 5

Dont put random stuff .................

Answers

Answer:

for the x+27

the value of X would be 76.5 bc the whole line is 180 degrees so you do x+27+x=180 then you solve

2x=153 then divide and you get 76.5

What is the area of this shape hurry

Answers

Answer:

15 units squared

Step-by-step explanation:

sorry if im wrong

Answer:

12

Step-by-step explanation:

Area of triangle + area of square + area of triangle = total area

(Area of a triangle: A = 1/2bh)

(Area of a square: A = bh)

Area of smallest triangle =

A = 1/2 (2) (2) = 2

Area of largest triangle=

A = 1/2 (2) (6) = 6

Area of the square =

A = 2 x 2 = 4

Total shape:

2 + 4 + 6 = 12

Emma is 14 years younger than twice fred age. Emma is 34. How old is fred

Answers

Answer:

24 years old

Step-by-step explanation:

34+14=48

48/2=24

Hope this helps =]

Answer:

82

Step-by-step explanation:

If Emma is 14 years younger then twice Fred's age, Then you add 7 to 34 with equals 41 then you want to do 41×2= which equals 82.

my math on that could be wrong and all you may have to do is add 34+14 but this kinda seems like a riddle thing but I hope I could help!!! ^-^

I NEED HEEEEEEELP-
But- People can you guys give me a straight answer? I'm tired and have a headache from all this. I need to make this assignment.
..........

Answers

Answer:

C= 93⁰, B= 64⁰, D= 23⁰

Step-by-step explanation:

180= (4x + 1) + (3x - 5) + x⁰

180= 4x+1 +3x-5 +x

180= 8x - 4

184= 8x

184/8= x

23=x

therefore

C= (4(23) +1)

C= 93

B= (3(23)-5)

B= 64

D= 23

93⁰+64⁰+23⁰=180⁰

In the data set {(7,7);(3,10);(5,9);(2,12);(0,1);(6,4);(4,10);(8,6)}, which point is a possible outlier?

(6,4)
(7,7)
(2,12)
(0,1)

Answers

6,4 and or 8,7 should be the answer

The possible outlier may be [tex](0,1)[/tex] as the second component is not greater than second components of the remaining pairs in comparison.

How to determine the possible outlier

In this data set, we observe that data set observes the rule that first component increases inasmuch the second component of the ordered pairs decreases. Thus, the possible outlier may be [tex](0,1)[/tex] as the second component is not greater than second components of the remaining pairs in comparison. [tex]\blacksquare[/tex]

To learn more on ordered pairs, we kindly invite to check this verified question: https://brainly.com/question/11740961

A farmhouse shelters 10 animals. Some are pigs and some are ducks. Altogether there are 36 legs. How many DUCKS are there?

Answers

Answer:

let no of pigs=x

ducks=y

x+y=10.....(1)

4x+2y=36

2x+y=18....(2)

by(1)..y=10-x

putting vvalue

2x+10-x=18

x=8

ducks=8

For problem 5, use the ordered pairs to answer the following question. SHOW YOUR WORK FOR FINDING SLOPE

Please can anyone give me the answer pleaseeeeeee

Answers

Answer:

what are the ordered pairs? then graph them and do the y axis number over the x axis number

Step-by-step explanation:

Solve the following inequality for w. Write your answer in simplest form
-w+9>6w+8

Answers

Answer:

-w+9>6w+8

-w+9 -8 >6w

-w+1>6w

1>7w

7w<1

w< 1/7

Step-by-step explanation:

Answer:

Isolate the variable by dividing each side by factors that don't contain the variable.

w< 1/7

or

(-∞,1/7)

Max picked out an old Star Wars puzzle to put together. Out of the 250 pieces, he was missing 15.
What percentage of the pieces was he missing?

Answers

Answer:

ummm... like 6% Convert fraction (ratio) 15 / 250 Answer: 6%

Step-by-step explanation: i have brain i use it some times

10) 3x + x + x + x - 3 - 2 = 7 + x + x
In the equation above, what is the value of x?

Answers

Answer:

Step-by-step explanation:

x=3

HELp Giving out brainlest and more points

Answers

Answer:

A

Step-by-step explanation:

Answer should be A

Powers multiply so

2*3=6

4*3= 12

(8^6)/(4^12)

Answer:

[tex]\frac{8^6}{4^{12}}[/tex]  A)

Step-by-step explanation:

[tex](\frac{8^2}{4^4} )^3\\\\\frac{8^6}{4^{12}}[/tex]

Exponent:

2 * 3 = 6

4 * 3 = 12

GIVING BRAINIEST IF CORRECT!!

Amanda slept for 1/3 of the day. She played with her friends for 1/5 of the day. She spent the rest of the day working on a school project. What fraction of the day did she spend working on her project?

Answers

Answer:

She spent 7/15 of the day working on her project.

Step-by-step explanation:

Well we cant get a clear answer with the fractions 1/3 and 1/5, so we are going to have to turn them into equivalent fractions.

3 and 5 both have 15 LCD so:

1/3 x 5/5 = 5/15

1/5 x 3/3 = 3/15

5/15 + 3/15 = 8/15

15/15 - 8/15 = 7/15

She spent 7/15 of the day working on her project.

Hopefully this helps :3 Sorry if wrong :( Plz mark brainiest if correct :D Your bootiful/handsome! Have a great day luv <3

-Bee~

Help I only have about 25 minutes left to finish this. I risk failing if I don't get this answered. I have 4 more questions after this. HELP PLZ. This is Slope Intercept form still. I will award 100 pts again

Answers

Answer:

y = 6x - 20

Step-by-step explanation:

Perpendicular slope = 6

(3, -2)

y - y1 = m(x - x1)

y + 2 = 6(x - 3)

y + 2 = 6x - 18

y = 6x - 20

Answer:

Solution given:

equation of a line is:

y=-⅛x+4.......[1]

Comparing above equation with y=mx+c

we get

slope [m]=-⅛

let slope of another line is M

since the Lines are perpendicular

so m ×M=-1

we get

M=8

again

the another line passes through(x1,y1) (3,-2)

Equation of another line becomes:

(y-y1)=m2(x-x1)

(y+2)=8(x-3)

y+2=8x-24

8x-y=24+2

8x-y=26

[tex] \frac{8x}{26} - \frac{y}{26} = 1[/tex]

[tex] \frac{4x}{13} - \frac{y}{26} = 1[/tex]

is a required equation of a line in slope intercept form.

Could someone send me answers, not some annoying link? I need help, not a virus.

Answers

Answer:

Equation: [tex]y=-\frac{1}{2} x+20[/tex]

20 represents the y-intercept. It is the initial value.

The slope is -1/2. It means that every gallon of gasoline used is used to shovel 2 driveways.

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