it is true that the expression (36⁴−6⁹)(38⁹−38⁸) is divisible by 30 and 37
Carl Friedrich Gauss introduced the fundamental principle of number theory in 1801, which states that any integer higher than one can be expressed as the product of prime numbers only in one way. Number theory is also referred to as arithmetic. Addition, subtraction, multiplication, and division are the four foundational operations in mathematics. Below, a quick discussion of all these operations is provided.
The expression is given as:
(36⁴−6⁹)(38⁹−38⁸)
Express 36 as 6²
(36⁴ - 6⁹) = (6²)⁴ - 6⁹
= 6⁸ - 6⁹
= 6⁷(6 - 6²)
= 6⁷(6 - 36)
= 6⁷(-30)
=(38⁹ - 38⁸)
= 38⁸(38 - 1)
= 38⁸(37)
=(36⁴−6⁹)(38⁹−38⁸)
=6⁷(-30) × 38⁸(37)
=(30)(37)(-6⁷)(38⁸)
Clearly, 30 and 37 are factors, so divisible by them.
The complete question is-
Prove that the value of the expression: b (36^5−6^9)(38^9−38^8) is divisible by 30 and 37.
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You need to give discharge instructions to the patient in room 408. The physician has prescribed an elixir that the patient will take at home. The order is for 1 ounce to be taken orally before bedtime.
How many teaspoons will the patient take?
Alternately, how many tablespoons will the patient take?
Using the method of conversion,
The patient will take 6 teaspoons of the elixir and 2 tablespoons of the elixir as prescribed by the physician.
What do you mean by conversion?Conversion means changing a particular unit of measurement into another so that the values or the measurements are in a single unit which then allows us to solve the problem or the sum easily. Both the base units compared should be same.
This helps to reduce the complications that are created in a question.
Now here in the question,
We have 1 ounce of elixir.
Now to convert 1 ounce in 1 teaspoon we need to multiply the value by 6.
Similarly in order to convert 1 ounce into tablespoon, we need to multiply it by 2.
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A circular garden has a circumference of 40.25 feet. Find the approximate diameter of the garden. Use 3.14 for π. Round to the nearest hundredth if necessary.
The approximate diameter of the garden is 12.82 feet.
What is circumference?
The circumference of a circle or ellipse in geometry is its perimeter. That is, if the circle were opened up and straightened out to a line segment, the circumference would be the length of the arc. The curve length around any closed figure is more often referred to as the perimeter.
Here, we have
Given: A circular garden has a circumference of 40.25 feet.
We have to find the diameter of the garden.
The formula for the circumference of a circle is given as 2πr.
We have to find the radius of the circular garden
Hence: 2πr = 40.25m
π = 3.14
To find the radius, the formula would be given as
r = C/2π
Where C = Circumference of the circle
r = (40.25/2)x 3.14
r = 63.195
d = C/π
d = 40.25/3.14
d = 12.82
Hence, the approximate diameter of the garden is 12.82 feet.
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Find the areas of the figures below. Can you find more than one method for each shape?
Please explain how you solved this. I am struggling.
Using area formula,
a. Area of the trapezium = 47.5m².
b. Area of parallelogram = 54inches².
Define area?The area of a thing is the amount of space it occupies in two dimensions. It is a way to count how many unit squares completely round the surface of a closed figure. The standard unit of area is the square unit, which is usually expressed as square inches, square feet, etc.
a. Sides of the trapezium are, a = 5m and b = 14m.
Height, h = 5m.
Area of the trapezium = (a+b)/2 × h
= (5 + 14)/2 × 5
= (19× 5)/2
= 95/2
= 47.5m².
b. Height of parallelogram = 6inches.
Base of parallelogram = 9inches.
Area = b × h
= 6 × 9
= 54inches².
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write your answer using only positive exponents
The value of the expression is 9y⁵z³/x⁶
How to determine the exponentNote that index forms are described as mathematical forms of writing numbers that are large or too small in more convenient forms.
The rules of index forms are;
Add the exponents when the values have the same base variable and are multiplying one another.Subtract the exponents when the values have the same base variable and are dividing one anotherFrom the information given, we have that;
(y³z⁻¹) (3x⁻³y/z⁻²)²
expand the bracket, we get;
(y³z⁻¹) 9x⁻⁶y²/z⁻⁴
Take the inverse exponent of the denominator
(y³z⁻¹) 9x⁻⁶y²z⁴
Now, add the exponents
9x⁻⁶y³⁺²z⁻¹⁺⁴
9x⁻⁶y⁵z³
Then, we have;
9y⁵z³/x⁶
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Triangle Trigonometry 2023 What is the value of x?
sin60=√3/2 = x/12
x=√3×12/2
x=√3×6
x=6√3
2. the new principal is $58,254.26. For payment 329, find the following:
a) The interest on $58,254.26
$
b) The payment to principal
$
c) The new principal
$
The interest on $58,254.26 was calculated to be $146.38, the payment to principal was calculated to be $182.62, and the new principal was calculated to be $58,071.64.
What is payment?Payment is the transfer of funds from one party to another in exchange for goods, services, or to fulfill a legal obligation. Payment can take various forms, including cash, check, credit card, money order, or electronic transfer. Payment can be made in person, over the phone, or online. The most important aspect of payment is that it is a legally binding contract between two parties. Payment is an important part of the economic system that facilitates the exchange of goods and services and helps to ensure that all parties involved receive what they expect from the transaction.
The interest on $58,254.26:
To calculate the interest on $58,254.26, we first need to determine the interest rate. Assuming that the interest rate is 3%, we can calculate the interest owed as follows:
Interest = Principal x Interest Rate x Time
Interest = $58,254.26 x 0.03 x 1/12
Interest = $146.38
b) The payment to principal:
To calculate the payment to principal, we need to subtract the interest from the total payment amount.
Payment to Principal = Total Payment – Interest
Payment to Principal = $329 - $146.38
Payment to Principal = $182.62
c) The new principal:
To calculate the new principal, we need to subtract the payment to principal from the original principal.
New Principal = Original Principal – Payment to Principal
New Principal = $58,254.26 - $182.62
New Principal = $58,071.64
In conclusion, the interest on $58,254.26 was calculated to be $146.38, the payment to principal was calculated to be $182.62, and the new principal was calculated to be $58,071.64.
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Ted invests in an annuity which earns 6.2% annual interest and he contributes $250 per month for 30 years. If Ted wants to perform some calculations to determine his future value, what formula would Ted use?
The future value of Ted's investment after 30 years would be approximately $13,711.86.
What is statistics?
Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.
Ted can use the formula for the future value of an annuity to calculate the future value of his investment:
FV = Pmt x (([tex](1 + r)^{n}[/tex]) - 1) / r
Where:
FV = Future Value
Pmt = Payment per period (monthly in this case)
r = Interest rate per period (monthly in this case)
n = Number of periods (in this case, 30 x 12 = 360 months)
Substituting the given values, we get:
FV = $250 x (([tex](1 + 0.062/12)^{360}[/tex] - 1) / (0.062/12)
FV = $250 x (283.8576) / 0.0051667
FV = $13,711.86
Therefore, the future value of Ted's investment after 30 years would be approximately $13,711.86.
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Select the correct answer.
Figure 1 is similar to figure 2. What is the height of figure 1?
4x
61-
figure 1
O A.
1 unit
OB.
O C.
O D. 4 units
8 units
2 units
-2x
figure 2
Answer:
D. 4 units
Step-by-step explanation:
Similar figures have corresponding lengths in proportion.
[tex]\frac{4x}{2x}=\frac{x+1}{x} \\ \\ 2=1+\frac{1}{x} \\ \\ 1=\frac{1}{x} \\ \\ x=1[/tex]
Therefore, the height of Figure 1 is 4.
Can someone give me the answers in order please
A car was valued at $44,000 in the year 1992. The value depreciated to $15,000 by the year 2006.
A) What was the annual rate of change between 1992 and 2006?
r=---------------Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r=---------------%
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2009 ?
value = $ -----------------Round to the nearest 50 dollars.
The rate of change between 1992 and 2006 is a decrease of 8.39%.
EquationsTo find the annual rate of change between 1992 and 2006, we can use the formula:
r = [tex](V2/V1)^{1/n}[/tex] - 1
where V1 is the initial value, V2 is the final value, and n is the number of years between the two values.
r = -0.0839
Therefore, the annual rate of change between 1992 and 2006 is -0.0839.
To express the rate of change in percentage form, we can multiply the result from part A by 100:
r = -0.0839 x 100
r = -8.39%
Therefore, the rate of change between 1992 and 2006 is a decrease of 8.39%.
To find the value of the car in the year 2009, we can assume that the value continues to drop at the same percentage rate as calculated in part A.
From 2006 to 2009, there are 3 years. So, using the formula for exponential decay, we have:
where V0 is the value in 2006, r is the rate of decrease, and n is the number of years between 2006 and 2009.
V = 11792.51
Therefore, the value of the car in the year 2009 would be approximately $11,800 (rounded to the nearest $50).
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Mr. Campbell has just purchased supplies for his classroom. He bought 76 markers, 31 rulers, and one projector. The markers cost 0.90 each, the rulers 1.60 each, and the projector 97.95 . What was the total amount that he spent?
Answer:
215.95
Step-by-step explanation:
68.4+49.6+97.95
I need help on this quickly!!
Answer:
Step-by-step explanation:
,mnlnnl;
Translate the argument to symbols and use a truth table to determine whether the argument is valid or invalid.
If it rains, then I will watch the Real World marathon.
It did not rain.
.. I did not watch the Real World marathon.
Let p="It rains," and let q="I will watch the Real World marathon."
Part: 0/4
Part 1 of 4
(a) Write the argument in symbols.
Translated into symbols:
If it rains, then I will watch the Real World marathon.
It did not rain.
.. I did not watch the Real World marathon.
The argument to symbols of the logic expression is shown below
Translating the argument to symbolsFrom the question, we have the following parameters that can be used in our computation:
If it rains, then I will watch the Real World marathon.It did not rain.I did not watch the Real World marathon.The argument in symbols can be expressed as:
p -> q
~p
Therefore, ~q
Where:
p = "It rains."
q = "I will watch the Real World marathon."
"->" denotes "implies".
"~" denotes "not".
So the argument can be read as "If it rains, then I will watch the Real World marathon. It did not rain.
Therefore, I did not watch the Real World marathon."
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which number line shows the solotion to one fourth r plus one < five
After looking at the provided number lines, it is clear that the number line (B) accurately reflects 1/4 + 1 < 5.
What is a number line?A number line is a diagram of a graduated straight line used to represent real numbers in introductory mathematics.
It is assumed that every point on a number line corresponds to a real number and that every real number corresponds to a point.
A number on the left of a number line is always smaller than a number on the right.
A number to the right is always greater than a number to the left, too.
So, the inequality we have is:
1/4 + 1 < 5
Now, after observing the given lines, we can easily tell that the number line (B) represents 1/4 + 1 < 5 perfectly.
Therefore, after looking at the provided number lines, it is clear that the number line (B) accurately reflects 1/4 + 1 < 5.
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Complete question:
Which number line shows the solution to 1/4 + 1 < 5
Jackson started a savings account with $25. He plans to deposit $25 each month for the next 12 months, then continue those monthly deposits in the following years. The account earns interest at an annual rate of 4% compounded annually, based on his final yearly balance. What is the total amouny that Jackson will save at the end of 3 years? How much intresr will jackson have earned at the end of the 3 years?
Jackson will have earned $72 in interest at the end of 3 years.
How to solve for the interest earnedJackson saves $25 per month for 12 months:
$25 * 12 months = $300 per year
Now, let's calculate the total amount saved at the end of each year:
Year 1: $300
Year 2: $300 + $300 = $600
Year 3: $300 + $600 = $900
Next, we'll apply the 4% interest rate compounded annually to the final yearly balance:
Year 1: $300 * (1 + 0.04) = $300 * 1.04 = $312
Year 2: $600 * (1 + 0.04) = $600 * 1.04 = $624
Year 3: $900 * (1 + 0.04) = $900 * 1.04 = $936
So, the total amount Jackson will save at the end of 3 years is:
$312 (Year 1) + $624 (Year 2) + $936 (Year 3) = $1,872
Total deposited:
$300 + $600 + $900
= $1,800
Total interest earned:
$1,872 - $1,800
= $72
So, Jackson will have earned $72 in interest at the end of 3 years.
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One square has a side 3 cm greater than the side of a second square. What are lengths of the sides if the sum of their areas is 1017 cm2. (Make sure to foil your product on the squares)
Using the area formula, the length of the sides of the two squares are 22 cm and 25 cm respectively.
What are squares?A square is a regular quadrilateral in Euclidean geometry, which means that it has four equal sides and four equal angles.
It can alternatively be explained as a rectangle with two neighboring sides that are of equal length.
The square's four sides are equal to one another or congruent.
The square's opposing sides run parallel to one another.
The square's diagonals form a 90° angle with one another.
The square's two diagonals are equal to one another.
Area: s²
s² + (s+3)² = 1017
s² + s² + 9 = 1017
2s² = 1017 - 9
2s² = 1,008
s² = 1008/2
s² = 504
s = ²504
s = 22.44
Round off: 22 cm
Then, (s+3) = 22 + 3 = 25 cm
Therefore using the area formula, the length of the sides of the two squares are 22 cm and 25 cm respectively.
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A textbook store sold a combined total of 440 math and history textbooks in a week. The number of math textbooks sold was three times the number of history textbooks sold. How many textbooks of each type were sold?
Number of math textbooks is 110 and number of history textbooks is 330 were sold when a textbook store sold a combined total of 440 math and history textbooks in a week.
There are two categories of textbooks. Let "x" stand for the quantity of math textbooks and "y" for the quantity of history textbooks.
A textbook store sold a combined total of 440 math and history textbooks in a week.
Equation,
x + y = 440
The number of math textbooks sold was three times the number of history textbooks sold.
x = 3y
Substituting the values in equation x + y = 440,
3y + y = 440
4y = 440
y = [tex]\frac{440}{4}[/tex]
y = 110
Number of math textbooks = 110
Number of history textbooks:
3y
3 × 110
330
Number of math textbooks is 110 and number of history textbooks is 330 were sold when a textbook store sold a combined total of 440 math and history textbooks in a week.
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savanna built a rectangular pen got her goat that was 20 yards long and 15.4 yards wide. what is the area of this pen?
Answer:
308 yards^2
Step-by-step explanation:
Area = length * width
Length = 20
Width = 15.4
20 * 15.4 = 308
Answer: 308 yards
Step-by-step explanation:
There are some sandwiches on the shelf.
Answer:
367
Step-by-step explanation:
Places in the alphabet:
T=20, H=8, E=5, R=18, E=5. 20+8+5+18+5=56. THERE=56
A=1, R=18, E=5. 1+18+5=24. ARE=24
S=19, O=15, M=13, E=5. 19+15+13+5=52. SOME=52
S=19, A=1, N=14, D=4, W=23, I=9, C=3, H=8, E=5, S=19. 19+1+14+4+23+9+3+8+5+19=105. SANDWICHES=105
O=15, N=14. 15+14=29. ON=29
T=20, H=8, E=5. 20+8+5=33. THE=33
S=19, H=8, E=5, L=12, F=6. 19+8+5+12+6=49. SHELF=49
56+24+52+19+105+29+33+49=367
HELP PLS THIS IS DUE TOMORROW PLSSSS HELP
Are ARST and ANSP similar? Use pencil and paper. Find the
measure of the angles in each triangle.
Which two angles must be congruent? What type of angles are they? (1 point)
Answer:
The angles are congruent, the two that are touching, m<s
Step-by-step explanation:
is this a right triangle
Given triangle is not a right angle triangle.
What is Pythagoras theorem?
For any right angle triangle,
Hypotenuse ²= Base²+Height ²
To determine whether the any triangle is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's denote the sides of the triangle as follows,
a = 14
b = 10
c = 4√19
Then we can check whether the Pythagorean theorem holds for these values:
c² = (4√19)² = 16×19 = 304
a²+ b² = 14²+ 10² = 196 + 100 = 296
Since c² is not equal to a² + b², the triangle is not a right triangle. Therefore, the answer is no, the given triangle is not a right angle triangle.
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rule 1
the output is the square of the input rule 2 the out put is 90 greater than the input
Rule 1: The output is the square of the input. For example, if the input is 3, then the output would be 9 (3 squared).
Rule 2: The output is 90 greater than the input. For example, if the input is 3, then the output would be 93 (3 + 90).
These are the two rules you've provided, and they describe how to calculate the output (y) based on the input (x).
Rule 1: The output is the square of the input.
In this rule, when you have an input value (let's call it x), you square it (multiply it by itself) to get the output value (y). Mathematically, this can be written as:
[tex]y = x^2[/tex]
Rule 2: The output is 90 greater than the input.
In this rule, when you have an input value (x), you add 90 to it to get the output value (y).
Mathematically, this can be written as:
y = x + 90.
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a company finds it can produce 30 heaters for $7500 while producing 40 heaters cost $9000, express the cost y as a linear function of the number of heaters x
express the cost y as a a linear function of the numbers of heatrs x
Answer:
y = 150x + 3000
Step-by-step explanation:
We can express the cost (y) as a linear function of heater quantity (x) by writing the equation in the slope-intercept form (y = mx + b), where m is the slope (in this case change in cost / change in heater quantity) and b is the y-intercept (when x = 0).
To find the slope (m), we can use the slope formula, which is
[tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex], where x1 and y1 are one point and x2 and y2 are the other point.
If we allow (30, 7500) to represent x1 and y1 and (40, 9000) to represent x2 and y2, our slope is:
[tex]m=\frac{9000-7500}{40-30}\\ \\m=\frac{1500}{10}\\ \\m=150[/tex]
Now, we can find the y-intercept by plugging in the slope and one of the points and solving for b:
[tex]7500=150(30)+b\\7500=4500+b\\3000=b[/tex]
Thus, our equation where y (cost) is a function of x (heater quantity) is y = 150x + 3000
If 9/23 = 0.3913043478260869565217, what is the 999th digit to the right of the decimal?
The 999th digit to the right of the decimal point is 9.
Describe Decimal Expansion?Decimal expansion is the representation of a number in decimal form. In decimal expansion, a number is represented using the base-ten positional notation system, where each digit represents a multiple of a power of ten.
For example, the decimal expansion of the number 1234.5678 is:
which simplifies to:
1000 + 200 + 30 + 4 + 0.5 + 0.06 + 0.007 + 0.0008 = 1234.5678
The decimal expansion of a number can be finite or infinite, and it may or may not be a repeating decimal. A repeating decimal is a decimal that has a repeating pattern of digits after the decimal point. For example, the decimal expansion of 1/3 is 0.3333..., with the digit 3 repeating infinitely.
To find the 999th digit to the right of the decimal point, we need to determine the repeating block of digits in the decimal expansion of 9/23.
The remainder when 9 is divided by 23 is 9, so we can start the long division as follows:
0.3913043478260869565217...
--------------------------------
23 | 9.00000000000000000000
- 6
--------
30
-23
-----
7
...
We can see that the decimal expansion of 9/23 repeats after 22 digits, since the remainder becomes 7, which is the same as the remainder we obtained after dividing 9 by 23.
Therefore, the 999th digit to the right of the decimal point is the same as the (999 mod 22)th digit, which is the 9th digit in the repeating block of digits:
0.3913043478260869565217...
9
So the 999th digit to the right of the decimal point is 9.
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What is the value of tan(-284°24') to the nearest ten-thousandth?
To find the value of tan(-284°24'), we can use the fact that the tangent function has period π, which means that:
tan(x) = tan(x + nπ)
where n is any integer. We can use this fact to convert the angle -284°24' to an equivalent angle between 0° and 360°:
-284°24' = -360° + 75°36' = 75°36'
Now, we need to find the reference angle, which is the acute angle between the terminal side of the angle and the x-axis. Since 75°36' is in the second quadrant (where the tangent function is positive), the reference angle is:
75°36' - 180° = -104°24'
Finally, we can use the identity:
tan(-θ) = -tan(θ)
to find the value of tan(-104°24'):
tan(-104°24') = -tan(104°24')
We can use a calculator to find that:
tan(104°24') ≈ 2.3835
Therefore:
tan(-284°24') ≈ -2.3835 (rounded to the nearest ten-thousandth)
So, the value of tan(-284°24') to the nearest ten-thousandth is -2.3835.
Three boxes are shipped on a truck. Each box has a base of 16 sq feet. Two boxes have a height of 3 feet and one box has a height of 5 feet. What is the total, volume, in cubic feet, of the three boxes?
Answer:
The total volume, in cubic feet, of the three boxes is 144 cubic feet.
Chain of thought:
1. Volume = base x height
2. Box 1: Volume = 16 sq feet x 3 feet = 48 cubic feet
3. Box 2: Volume = 16 sq feet x 3 feet = 48 cubic feet
4. Box 3: Volume = 16 sq feet x 5 feet = 80 cubic feet
5. Total Volume of three boxes = 48 + 48 + 80 = 144 cubic feet
How many 5-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if repetitions of digits are allowed?
The 5-digit number that can be formed is 100,000.
5-digit numbers using 0,1,,2,3,4,5,6,7,8,9,
We can use any of the 10 numbers to form a 5-digit number
The first place, there can be any of the 10 numbers
In the second place, there can be any of the 10 numbers
In the third place, also there can be any of the 10 numbers and so on.
Therefore we have:
10×10×10×10×10 = 100,000 digits
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Suppose a shipment of 400 components contains 68 defective and 332 non-defective computer components. From the shipment you take a random sample of 25. When sampling with replacement (so that the p = probability of success does not change), note that a success in this case is selecting a defective part. What is the probability of there being no defects in the entire shipment?
The probability of there being no defects in the entire shipment is 0.2075.
What is probability?
Probability can be used to calculate an event's probability. It can only be used to assess the possibility that an event will occur. A scale from 0 to 1, where 0 is impossible and 1 is a particular occurrence.
We are given that out of the 400 components, 68 are defective and 332 are non-defective computer components.
A random sample of 25 is taken.
So,
⇒ Probability (Defects in the shipment) = [tex]\frac{68}{400}[/tex]
⇒ Probability (Defects in the shipment) = 0.17
Thus,
⇒ Probability (No Defects in the shipment) =25 * (1 - 0.17)
⇒ Probability ( No Defects in the shipment) = 25 * (0.83)
⇒ Probability ( No Defects in the shipment) = 0.2075
Hence, the probability of there being no defects in the entire shipment is 0.2075.
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The ratio of the number of men to the number of women was 5:9 at a museum. The ratio became 2:3 when 42 women left the museum. Find the total number of men and women at the museum at the beginning.
Answer:
150 men and 252 women
Step-by-step explanation:
Ratio:Men : Women = 5 : 9
Number of men = 5x
Number of women = 9x
42 women left the museum.
Number of remaining women = 9x - 42
New ratio = 2 : 3
[tex]\dfrac{5x}{9x -42}=\dfrac{2}{3}\\\\\\\bf Cross \ multiply,[/tex]
3*5x = 2*(9x - 42)
15x = 2*9x - 2*42
15x = 18x - 84
18x - 84 = 15x
Add 84 to both the sides,
18x = 15x + 84
Subtract 15x from both sides,
18x - 15x = 84
3x = 84
Divide both sides by 3,
x = 84 ÷ 3
x = 28
Number of men = 5 * 28 = 140
Number of women = 9*28 = 252
Which of the following are the next three terms in the sequence (1, 2, 3, 9, -27, 81, ..}?
The next three terms are:
-243, 729, -2187help mee with this question
Answer: 22
Step-by-step explanation:
The first one is 10.
The other 2 has 6, because 10-4=6
So
10+(6x2)=22