3²×5, 2⁴×5, 2²×3×5×7, 5⁴ and 3⁴ are the index notations of the numbers 45, 80, 420, 625 and 81 respectively. This can be obtained by taking the factors of these numbers and wring as the powers as shown below.
What is index notation?
It is a method of representing numbers and variables that are multiplied by themselves a number of times.
For example, 7³, a²
What are the index notations of the given numbers?
The given numbers can be represented as the following:
(1) 45 =3×3×5 = 3²×5
(2) 80 =2×2×2×2×5 = 2⁴×5
(3) 420 =2×2×3×5×7 = 2²×3×5×7
(4) 625 =5×5×5×5 = 5⁴
(5) 81 =3×3×3×3 = 3⁴
Hence 3²×5, 2⁴×5, 2²×3×5×7, 5⁴ and 3⁴ are the index notations of the numbers 45, 80, 420, 625 and 81 respectively.
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the question is in the attached picture
Answer: [tex]\frac{1}{3} \sin x^{3}+C[/tex]
Step-by-step explanation:
Let [tex]u=x^3[/tex]. Then, [tex]3x^2 dx = du \longrightarrow x^2 dx =\frac{1}{3}du[/tex]
So, we can rewrite the original integral as
[tex]\frac{1}{3} \int \cos u \text{ } du=\frac{1}{3} \sin u+C=\frac{1}{3} \sin x^{3}+C[/tex]
MATH HELP!!! 100PTS!!!
Suppose that u=log10(3) and v=log10(5). Find possible formulas for the following expressions in terms of u and/or v. Your answers should NOT involve any log 's.
a) log10(0.6)=
u-v
b) log10(0.25)=
c) log10(27000)=
3+3u
d) log10(sqrt10)=
Answer:
a) u - v
b) 2v - 2
c) 3u + 3
d) ¹/₂
Step-by-step explanation:
Given:
[tex]u=\log_{10}3[/tex]
[tex]v=\log_{10}5[/tex]
Part (a)
Rewrite 0.6 as a fraction:
[tex]\implies \log_{10}(0.6)=\log_{10}\left(\dfrac{3}{5}\right)[/tex]
[tex]\textsf{Apply the quotient log law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay:[/tex]
[tex]\implies \log_{10}\left(\dfrac{3}{5}\right)=\log_{10}3-\log_{10}5[/tex]
Substitute the values of u and v:
[tex]\implies \log_{10}3-\log_{10}5=u-v[/tex]
Part (b)
Rewrite 0.25 as 25/100:
[tex]\implies \log_{10}(0.25)=\log_{10}\left(\dfrac{25}{100}\right)[/tex]
[tex]\textsf{Apply the quotient log law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay[/tex]
[tex]\implies \log_{10}\left(\dfrac{25}{100}\right)=\log_{10}(25)-\log_{10}(100)[/tex]
Rewrite 25 as 5² and 100 as 10²:
[tex]\implies \log_{10}(25)-\log_{10}(100)=\log_{10}(5^2)-\log_{10}(10^2)[/tex]
[tex]\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies \log_{10}(5^2)-\log_{10}(10^2)=2\log_{10}5-2\log_{10}10[/tex]
[tex]\textsf{Apply the log law}: \quad \log_aa=1[/tex]
[tex]\implies 2\log_{10}5-2\log_{10}10=2\log_{10}5-2(1)[/tex]
Substitute the value of v:
[tex]\implies 2\log_{10}5-2(1)=2v-2[/tex]
Part (c)
Rewrite 27000 as 30³:
[tex]\implies \log_{10}(27000)=\log_{10}(30^3)[/tex]
[tex]\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies \log_{10}(30^3)=3\log_{10}(30)[/tex]
[tex]\textsf{Apply the log product law}: \quad \log_axy=\log_ax + \log_ay[/tex]
[tex]\implies 3\log_{10}(30)=3\log_{10}(3)+3\log_{10}(10)[/tex]
[tex]\textsf{Apply the log law}: \quad \log_aa=1[/tex]
[tex]\implies 3\log_{10}(3)+3\log_{10}(10)=3\log_{10}(3)+3(1)[/tex]
Substitute the value of u:
[tex]\implies 3\log_{10}(3)+3(1)=3u+3[/tex]
Part (d)
Rewrite √10 as [tex]10^{\frac{1}{2}}[/tex] :
[tex]\implies \log_{10}(\sqrt{10})=\log_{10}(10^{\frac{1}{2}})[/tex]
[tex]\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies \log_{10}(10^{\frac{1}{2}})=\dfrac{1}{2}\log_{10}(10)[/tex]
[tex]\textsf{Apply the log law}: \quad \log_aa=1[/tex]
[tex]\implies \dfrac{1}{2}\log_{10}(10)=\dfrac{1}{2}(1)=\dfrac{1}{2}[/tex]
What other circle theorems can cyclic quadrilaterals can be related to? Explain how.
The circle theoremss that cyclic quadrilaterals can be related to include:
Every corner of the quadrilateral touches the circumference of the circle. The ratio between the diagonals and the sides can be defined.The product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.What is a cyclic quadrilateral?It should be noted that cyclic quadrilateral simply means a quadrilateral that's drawn inside a circle.
In this case, every corner of the quadrilateral must touch the circumference of the circle.
Also, the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.
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In a study to add a new feature to a software program, the programmer introduced two categories, men and women, in the survey she conducted. Is the study observational or experimental? if it is an experiment, what is the controlled factor?.
The is an observational study.
What is an observational study?
In an observational study, researchers examine the effects of a particular intervention, risk, diagnostic test, or treatment without trying to control who is exposed to it.
By having a control group, or those who are not exposed, and an experimental group, or those who are subjected to the intervention, treatment, etc., an observational study contrasts from an experimental research, where the scientists are controlling who gets exposed to the treatment, intervention, etc. The groups are randomly assigned or picked in the best studies.
Reason Behind it Being an Observational Study
Since, in this case also, the two categories of men and women, the candidates are randomly picked for the survey of the new software programmer feature. Hence, it is a scenario of observational study.
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Determine the equation, in both factored and standard forms of the following parabola
The standard form of the equation of the parabola is y + 3 = (1/3) · (x - 3)² and The factored form of the equation of the parabola is y = (1/3) · x · (x - 6).
How to derive the two forms of the equation of the parabola
Mathematically speaking, parabolae are defined by second order polynomials. The standard form of the equation of the parabola is defined by:
y - k = C · (x - h)² (1)
Where C is the vertex constant.
And the factored form has this form:
y = k · (x - r₁) · (x - r₂) (2)
First, we determine the standard form of the equation of the parabola: (h, k) = (3, - 3) and (x, y) = (0, 0)
0 - (- 3) = C · (0 - 3)²
3 = 9 · C
C = 1/3
Then, the standard form of the equation of the parabola is y + 3 = (1/3) · (x - 3)². Now we proceed to determine the factored form of the parabola: (r₁ = 0, r₂ = 6), (h, k) = (3, - 3)
- 3 = k · (3 - 0) · (3 - 6)
- 3 = k · 3 · (- 3)
- 3 = - 9 · k
k = 1/3
The factored form of the equation of the parabola is y = (1/3) · x · (x - 6).
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Please help ill give whoever answers the best the brainliest answer :0
Expression (5x² - 2x + 7), ([tex]\frac{1}{5}x^3 -\frac{7}{2}x^2 -2[/tex]) and (y⁵ - √(5y²) + 7) are polynomials.
What is a polynomial ?A polynomial is a type of algebraic expression where exponents of all variables are whole number.
1) 5x² - 2x + 7
Since there are no variables raised to negative or fraction exponents,
the expression is a polynomial.
2) [tex]\frac{1}{5}x^3 -\frac{7}{2}x^2 -2[/tex]
Since there are no variables raised to negative or fraction exponents,
the expression is a polynomial.
3) y⁵ - √(5y²) + 7
This can be re-written as
[tex]y^5-(\sqrt{5})y+7[/tex]
Since there are no variables raised to negative or fraction exponents,
the expression is a polynomial.
4) [tex]\frac{y^3}{3} -\sqrt{y} +2[/tex]
This can be re-written as
[tex]\frac{1}{3}y^2 -y^{1/2}+2[/tex]
Since variable y is raised to a fraction exponent,
the expression is not polynomial.
Therefore, expression (5x² - 2x + 7), ([tex]\frac{1}{5}x^3 -\frac{7}{2}x^2 -2[/tex]) and (y⁵ - √(5y²) + 7) are polynomials.
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PLEASE HELP I NEED THIS SOON! The monthly revenue of a digital magazine from each user is represented by the function f(x) = 4x + 1. The monthly cost to provide the digital magazine to each user per month is represented by the function g(x) = x2. Which of the following evaluates the profit function for each user of the digital magazine after 2 months?
A (f • g)(2) = (4(2) + 1)(22) = (9)(4) = $36
B (f + g)(2) = (4(2) + 1) + 22 = 9 + 4 = $13
C (g − f)(2) = 22 − (4(2) + 1) = 4 − 9 = −$5
D (f − g)(2) = (4(2) + 1) − 22 = 9 − 4 = $5
Answer:
The answer is D(f-g)
Step-by-step explanation:
Profit minus expenses
Answer:
D
Step-by-step explanation:
The profit can be defined as (cost of production) - (revenue) and since the cost is defined as g(x) = x^2 and the revenue is f(x) = 4x+1. The profit can be defined as [tex](f-g) = (4x+1)-x^2 = -x^2+4x+1[/tex]. After 2 months you can calculate the profit by simply plugging in 2 as x. This gives you the equation: [tex](f-g) = (4(2) + 1) - 2^2 = 9-4 = 5\$[/tex]
Joan draws a bar chart to show the temperatures at midday on March 1st in five cities.
Joan has made four mistakes in her diagram.
Write down any three of them.
The mistakes that Joan made are:
The y-axis does not start from zero.The width of the bars are not equal.The space between the bars are not equal.What is a bar chart?A bar chart is a graph that shows the numerical values of variables represented by the height rectangles of equal width.
In a bar graph, the rectangles must be of equal width and the space between each bars must be equal.
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The slope of the line below is 3. Use the coordinates of the labeled point to
find a point-slope equation of the line.
A. y-6-3(x-8)
B. y-8--3(x-6)
C. y-6--3(x-8)
D. y-8-3(x-6)
10
(6,8)
10
The point-slope equation of the line is y - 8 = 3(x - 6)
How to determine the equation?The points are given as:
(x1, y1) = (6, 8)
The slope is given as:
m= 3
The equation is then calculated as:
y - y1 = m(x - x1)
This gives
y - 8 = 3(x - 6)
Hence, the point-slope equation of the line is y - 8 = 3(x - 6)
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Juma spent half of his July
The total salary juma receives for the month of July before the deductions is sh. 12,800
Amount of earningsLet
Total salary = xAmount spent on school fees = 1/2xAmount spent on farming = 1/8xSchool fees + farming = 1/2x + 1/8x
= 4x+x/8
= 5/8x
Amount remaining = x - 5/8x
=8x-5x /8
= 3/8x
Amount spent on food = 2/3 × 3/8x
= 6/24x
= 1/4x
So,
1/4x = sh.3200
x = 3200 ÷ 1/4
x = 3200 × 4
x = sh. 12,800
Complete question:
Juma spent half of his salary on school fees one-eighth on farming and two thirds of the reminder on food . Calculate his July salary if he spent sh.3200 on food
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A square piece of card has a square of side 2 cm cut out from
each of its corners. The remaining card is then folded along
the dotted lines shown to form an open box whose total
internal surface area is 180 cm².
What is the volume of the open box in cm³?
A 100 B 128 C 162 D 180 E 200
Answer:
E) 200
Step-by-step explanation:
Let the length of the side after cutting the corners =x cm
Length of the side before cutting the corners = x + 4 cm
Area of square before cutting = (x + 4)² square cm
Area of that square that was cut = 2*2 = 4 cm²
Area of 4 squares cut from all 4 corners = 4 * 4 = 16 cm²
Area of internal surface area = 180 square cm
(x + 4)² - 4*4 = 180
x² + 2*4*x + 4² - 16 = 180
{expand using the identity (a + b)² = a² + 2ab + b²) }
x² + 8x + 16 -16 = 180
x² + 8x - 180 = 0
x² + 18x - 10x - 180 = 0
x( x + 18) - 10(x + 18) = 0
(x + 18) (x - 10) = 0
x - 10 = 0 {Ignore x +18 = 0}
x = 10
Open box:length = x = 10 cm
breadth = x = 10 cm
height = 2 cm
Volume of open box = length * breadth * height
= 10 * 10 * 2
= 200 cm³
What is the quotient of startfraction negative 8 x superscript 6 baseline over 4 x superscript negative 3 baseline endfraction?
The fractional number given has a quotient of -2x⁹.
Given that (-8x⁶)÷(4x⁻³) is the fractional number.
The resultant number obtained by dividing two numbers is known as the quotient. Let's divide number a by number b. Then, q=a÷b will be these two numbers' quotient.
Here are the actual numbers (a,b).
In this case,the fraction is (-8x⁶)÷(4x⁻³).
Assume that q=(-8x⁶)÷(4x⁻³)
If you write the denominator variable as x⁻¹=1÷x, you obtain
[tex]\begin{aligned}q&=\frac{-8x^6}{4\frac{1}{x^3}}\\ &=\frac{-8x^6\times x^3}{4}\end[/tex]
As we are aware, if two numbers are multiplied together and their bases are the same, their powers can be represented as sums, such as
q=(-8x⁶⁺³)÷(4)
q=(-8x⁹)÷(4)
By dividing the numerator by the denominator, we get
q=-2x⁹
Hence,the given fraction (-8x⁶)÷(4x⁻³) has a quotient of -2x⁹.
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Answer: option D -2x^9 on edge 2023
Step-by-step explanation:
If diana walks forward and her angle looking to the top of the building changes to 40°, how much closer is she to the building? round the answer to the nearest tenth of a foot. 10.3 ft 17.6 ft 30.2 ft 97.2 ft
Answer:
17.6 ft
Step-by-step explanation:
tan 37 = 130/x
x = 130/tan 37 = 130/0.7536 = 172.5 feet
tan 40 = 130/y
y = 130/tan 40 = 130/0.8391 = 154.9
172.5 - 154.9 = 17.6 ft
Answer:
17.6 Ft
Step-by-step explanation:
Geometry B E2020!! :3
Are these two angles corresponding?
Answer:
Yes they are
Step-by-step explanation:
Because AE and FD both make a 50 degree angle with AC they are parallel which means that the two angles are corresponding.
40 MORE POIMNTSSSSSSSSSSSSSSSSS
[tex]y = \sqrt[3]{x + 10} [/tex]
[tex]x = \sqrt[3]{y + 10} [/tex]
[tex]x {}^{3} = y + 10[/tex]
[tex]y = {x}^{3} - 10[/tex]
Letter B# [4-5{1-(4-3+1)}] of 100% equals
a) 50%
b)-1%
c)zero
d)100%
Answer:
To solve this, we need to first evaluate the given expression. We can do that by simplifying the contents of each bracket, starting with the innermost one:
[4-5{1-(4-3+1)}]
⇒ [4 - 5{1 - (2)}]
⇒ [4 - 5{-1}]
⇒ 4 - (-5)
⇒ 4 + 5
⇒ 9
Now we can calculate what 9 out of 100 is:
9 of 100 = [tex]\frac{9}{100}[/tex]
⇒ 9%
if the reserve ratio is 50%, and $10,000 in new deposits is added to the banking system, how much of the new deposits is the bank required to keep on hand? (eco class
$5000
$500
$1500
$50
Step-by-step explanation:
5,000 by dividing 10.000
$5000 of the new deposits is the bank required to keep on hand.
The answer is option A.
What is problem-solving?Problem-solving is the act of defining a problem; figuring out the purpose of the trouble; identifying, prioritizing, and selecting alternatives for an answer; and imposing an answer.
Problem-solving starts with identifying the issue. As an example, a trainer may need to parent out a way to improve a scholar's overall performance on a writing talent test. To do that, the instructor will overview the writing exams looking for regions for improvement.
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Type the correct answer in each box. Round your answers to the nearest integer.
-10
10
-10
In the figure, the perimeter of hexagon ABCDEF is approximately
Reset
A
10
B
15 20
Next
D
units, and its area is
C
square units.
Answer:
500 is your answer.
Step-by-step explanation:
First solve the length of side BC, CD, EF and FA Since BC = CD = sqrt( 10^2 + 10^2) BC = CD = 14.1421
FA = EF = sqrt(10^2 + 20^2) = 23.3607
So the perimeter = 10 + 10 + 14.1421 + 14.1421 + 23.3607 = 93
The area is made up be triangle FAE, rectangle ABDE and triangle BCD
A = 0.5(20)(20) + (10)(20) + 0.5(20)(10)
A = 500 sq units
A company makes 352 boxes of cereal each day. how many boxes are made in 7 days?
Answer: 2,464 boxes are made in 7 days.
Step-by-step explanation:
1 day = 352 boxes
7 days = ?
To solve this word problem, you can multiply 352 and 7.
352*7= 2464
find the perimeter of each of the rectangles whose sides are given as follows a. 5.5cm and 4.8 cm
Answer:
here is what I came up with 26.4
i need help fast please
Answer:
The answer is 540 mm^3
Step-by-step explanation:
using v= BH
v= 30×18
v= 540mm^3
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does this table represent a function ? why or why not
Answer:
yes
Step-by-step explanation:
every hours of training has mapped to monthly pays and each element in hour of training has mapped to unique element in monthly pay.
5. 61°N, 150°W 5. 61 ° N , 150 ° W
Based on the coordinates given, the location that is being described is Cook Inlet, in the Gulf of Alaska.
Where do the given coordinate lead to?The coordinates of 61°N, 150°W means that the place is located in the Northern and Eastern hemisphere.
When a map is used, it points to an area in the Gulf of Alaska that is very close to the Cook Inlet which is close to Anchorage.
Question is:
Determine what country or place is located on the given approximate coordinates.
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Solve each of the following by using the general quadratic formula: (2 decimal places)
The solutions to the given quadratic equation [2x² - 3x - 4 = 0] are x = 2.35, -8.85.
The solutions to the given quadratic equation [x² + 2x + 2 = 0] are x = -1+i, -1-i.
What are the solutions to the quadratic equation?Given the equation for the general quadratic formular;
[ -b±√( b² - 4(ac)) ]/2a
a) 2x² - 3x - 4 = 0
a = 2b = -3c = -4We substitute into the equation.
x = [ -(-3)±√(-3)² - 4( 2 × -4 )) ] / 2×2
x = [ 3±√9 + 32 ] / 4
x = [ 3±√41 ] / 4
x = [ 3+√41 ] / 4, [ 3-√41 ] / 4
x = 2.35, -8.85
The solutions to the given quadratic equation [2x² - 3x - 4 = 0] are x = 2.35, -8.85
b) x² + 2x + 2 = 0
a = 1b = 2c = 2We substitute into the equation.
x = [ -2±√( 2² - 4( 1×2 )) ] / 2×1
x = [ -2±√( 4 - 8) ] / 2
x = [ -2±√(-4) ] / 2
Lets re-write -4 as -1( 4 )
x = [ -2±√( -1 × 4) ] / 2
x = [ -2±√4 × √-1 ] / 2
Write √-1 as i
x = [ -2±√4 × i] / 2
x = [ -2± 2 × i] / 2
x = 2[ -1 ± i] / 2
x = (-1 ± i)
x = -1+i, -1-i
The solutions to the given quadratic equation [x² + 2x + 2 = 0] are x = -1+i, -1-i.
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A function g(x) has x-intercepts at (startfraction 1 over 2 endfraction, 0) and (6, 0). which could be g(x)?
If function has points (1/2,0) and (6,0) then [tex]g(x)=2x^{2} -13x+6\\[/tex]
Given x-intercepts at (1/2,0)(6,0) and we have to find the function g(x)
If the x-intercepts lie at (1/2,0)(6,0) then the function has roots of 6 and 1/2. Function is a relationship between variables having value of y for each value of x. Value of x is domain and value of y is known as codomain.
If we insert x=6 or x=1/2 into the equation of g(x) we will obtain 0 as under:
g(6)=0
In order to get 0 the equation will be x-6.
In order to get 0 for root the equation will be x-1/2.
but x-1/2 does not appear in any of these but its multiples can be there :
[tex]2(x-1/2)=2x-1[/tex]
Therefore the function will be as under:
[tex]g(x)=(x-6)(2x-1)[/tex]
[tex]g(x)=2x^{2} -x-12x+6[/tex]
[tex]g(x)=2x^{2} -13x+6[/tex]
Hence the function g(x) is [tex]g(x)=2x^{2} -13x+6[/tex].
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please help me im in a rush
Answer:
[tex] \frac{x}{y} = \frac{10}{5} \\ = \frac{5}{1} [/tex]
1. •°• y is 5 times x
[tex] \frac{x}{y} = \frac{30}{5} \\ = \frac{6}{1} [/tex]
2. •°• y is 6 times x
[tex] \frac{x}{y} = \frac{56}{8} \\ = \frac{7}{1} [/tex]
3. •°• y is 7 times x
[tex] \frac{x}{y} = \frac{88}{11} \\ = \frac{8}{1} [/tex]
4. •°• y is 8 times x
[tex]2 \times \frac{7}{4} [/tex]
•°• y is 2 times x
[tex]5 \times \frac{7}{4} [/tex]
•°• y is 5 times x
[tex]8 \times \frac{7}{4} [/tex]
•°• y is 8 times x
[tex]11 \times \frac{7}{4} [/tex]
•°• y is 11 times x
Please help question in the picture
Answer:
D
Step-by-step explanation:
D is correct because the sqrt of 91 is 9.5 and 9<9.5<10 makes absolute sense
C is not correct because the sqrt of 33 is 5.7 and 6<5.7<7 does not make sense
B is not correct because the sqrt of 13 is 3.6 and 4<3.6<5 does not make sense
A is not correct because the sqrt of 65 is 8.06 and 7<8.06<8 does not make sense
Can you answer my question please and thank you
Answer:
b = 12
Step-by-step explanation:
since a² + b² = c²
we can find b by pythagorean theorem.
sqrt(13² - 5²) = b
hence,
b = 12
Answer: 12
Step-by-step explanation:
To find a length of a right triangle, we can use Pythagorean theory:
leg² + leg² = hypotenuse²
5² + b² = 13²
25 + b² = 169
b² = 144
b = [tex]\sqrt{144}[/tex]
b = 12
A line passes through the point (8, -9) and had a slope of -5/4. Write an equation in slope-intercept form for this line.
Answer:
y = -5/4x +1
Step-by-step explanation:
The slope-intercept form of a line is given by
y = mx + b where m is the slope and b is the y intercept
We are given the slope and a point on the line
y = -5/4 x + b
We can substitute the point for x and y and solve for b
-9 = -5/4(8)+b
-9 = -10+b
Add 10 to each side
-9+10=-10+10+b
1 = b
The slope intercept form of the equation is
y = -5/4x +1
A LANE IN THE MIDDLE OF A TWO-WAY ROAD THAT HAS DASHED LINES ON BOTH SIDES AND TWO LARGE CURVED ARROWS FACING EACH OTHER MAY BE USED FOR:
This lane symbol is used for marking the beginning and ending of a left turn to the drivers.
What are lane symbols?
A permitted lane usage is denoted by a lane symbol. For example, a lane marked with a diamond is one that is designated for high-occupancy cars. A lane symbol of a bicycle marks that the lane is designated for cyclists. At crossings, arrows indicate movements that are necessary or allowed. The user of the roadway must yield when there is a row of solid triangles. Likewise, two large curved arrows facing each other on a lane in the middle of a two way road with dashed lines on either sides indicate the beginning and ending of a left turn.
A Lane symbol or marking are also utilized to warn drivers of potentially dangerous situations that may be coming up. For instance, a triangle with no filling indicates a yield ahead. A speed hump is indicated by a succession of lines across a lane that get bigger and wider.
Learn more about a lane symbol here:
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