1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
8_murik_8 [283]
3 years ago
11

How many different numbers between $\dfrac{1}{1000}$ and $1000$ can be written either as a power of $2$ or as a power of $3$, wh

ere the exponent is an integer?
Mathematics
2 answers:
Pani-rosa [81]3 years ago
7 0

Answer:

31

Step-by-step explanation:

Sever21 [200]3 years ago
3 0

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.

You might be interested in
Evaluate in the following expression when x = 4 and y = 5​
PIT_PIT [208]
Fraction with x squared plus y cubed in the numerator and 2 plus x in the denominator.
3.8
14.83
19.58
23.5
7 0
2 years ago
Quiz Questions
gavmur [86]
1. 6:3
2. 6:2
3. 3:18
4:5:6
5:13
6:x
7:x
8: check
9:check
10: x
6 0
2 years ago
3. Give the slope and y-intercept of the<br> relationship shown in the table.
kolezko [41]

Answer:

B

Step-by-step explanation:

Since the slope is going down the slope is negative and since 50 is positive when 0 its positive.

Also to find slope you do y/x

25/100=1/4 and since its a decreasing pattern the slope is -1/4

4 0
3 years ago
Find the equation of a parallel to y+2=-1/2x that passes through the point (-2,2)
FromTheMoon [43]

Answer:

y = - 1/2x + 1

Step-by-step explanation:

y+2= - 1/2x

Slope = -1/2

Point: (-2,2)

y-Intercept: 2 - (-1/2)(-2) = 2 - 1 = 1

4 0
2 years ago
X+3y=3<br> 2x-4y=6<br><br> solve the equations
Gelneren [198K]
Subtract the second equation from double the first equation:

2x+6y-2x-(-4y)=0

10y=0

Therefore, y=0.  Substituting this into the first equation, we see that:

x+3(0)=3

x=3

Therefore, x=3.  The solution to this system is x=3, y=0, or (3, 0).
4 0
3 years ago
Read 2 more answers
Other questions:
  • The slope-intercept form of the equation of a line that passes through point (-3, 8) is y=-2/3x+6. What is the point-slope form
    7·1 answer
  • During summer, when the snow melts, the water level of the Yukon River increases. If the water level of the river increased by 2
    5·1 answer
  • What number must be added to14,056 to result the sum of 38,776
    11·1 answer
  • What is the solution of 3|x – 5| + 2 = 20?
    10·2 answers
  • Increase £500 by 3.2% how do you work it out
    13·1 answer
  • Please help.I really don’t understand this..
    12·2 answers
  • Help please &amp; thank you !!
    8·1 answer
  • A swimming pool in the shape of a rectangular prism is filled with water. The pool measures 12 ft wide by 24 ft long by 6ft deep
    10·1 answer
  • I need help please.
    10·1 answer
  • Sarah jogged 4.8 miles each day for 22 days last month. How many miles did Sarah jog in total?
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!