Explanation: Zero Exponent
So far we have looked at powers of 10 with exponents greater than 0.
What would happen to our patterns if we included 0 as a possible exponent?
1. Write 10^12 × 10^0 with a power of 10 with a single exponent using the appropriate exponent rule. Explain or show your reasoning.
A. What number could you multiply 10^12 by to get this same answer?
Answers
Answer:
1) We can use the following property to simplify the product of powers as follows:
\(a^{m}\cdot a^{n} = a^{m + n}\), \(a\in \mathbb{R}\), \(m,n\in \mathbb{Z}\) (1)
Therefore, we have the following result:
\(10^{12}\cdot 10^{0} = 10^{12+0} = 10^{12}\)
A) \(10^{12}\) must be multiplied either by \(10^{0}\) or by \(1\) to get the same answer.
Step-by-step explanation:
1) We can use the following property to simplify the product of powers as follows:
\(a^{m}\cdot a^{n} = a^{m + n}\), \(a\in \mathbb{R}\), \(m,n\in \mathbb{Z}\) (1)
Therefore, we have the following result:
\(10^{12}\cdot 10^{0} = 10^{12+0} = 10^{12}\)
A) In addition, we can use this property:
\(\frac{a^{m}}{a^{n}} = a^{m-n}\), \(a\in \mathbb{R}\), \(m,n\in \mathbb{Z}\)
We can apply the property mentioned above:
\(10^{12}\cdot 10^{0} = 10^{12}\cdot (10^{n-n}) = 10^{12}\cdot \left(\frac{10^{n}}{10^{n}} \right) = 10^{12} \cdot 1 = 10^{12}\)
In consequence, we conclude that \(10^{0} = 1\). \(10^{12}\) must be multiplied either by \(10^{0}\) or by \(1\) to get the same answer.
Related Questions
are the following statements true or false? false 1. if two row interchanges are made in sucession, then the determinant of the new matrix is equal to the determinant of the original matrix. false 2. if is zero, then two rows or two columns are the same, or a row or a column is zero. false 3. the determinant of is the product of the diagonal entries in . false 4. .
Answers
Statements are False: 1. interchanging rows changes determinant. 2. determinant zero not implies specific rows. 3. determinant not product of diagonal entries. 4. determinant is scalar not matrix.
What is matrix ?
A matrix is a rectangular array of numbers or other mathematical objects, typically arranged in rows and columns. Matrices are often denoted using capital letters, such as A, B, and C. Each element of a matrix is identified by its row and column indices,
1) False, If two row interchanges are made in succession, the determinant of the new matrix is the negative of the determinant of the original matrix.
2) False, If the determinant of a matrix is zero, it does not necessarily mean that two rows or two columns are the same or a row or column is zero, it only means that the matrix is singular, i.e. non-invertible and it also can mean that the matrix is linearly dependent.
3) False, The determinant of a matrix is not always equal to the product of the diagonal entries, it is a scalar value calculated through a specific method called matrix expansion which is based on the entries of the matrix and it depends on the size of the matrix.
4) False, The determinant of a matrix is a scalar value, it cannot be equal to another matrix.
Statements are False: 1. interchanging rows changes determinant. 2. determinant zero not implies specific rows. 3. determinant not product of diagonal entries. 4. determinant is scalar not matrix.
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Find the indicated term of the binomial expansion. 14th; (c-2)15 What is the 14th term? -860,160c² (Simplify your answer.)
Answers
The 14th term of the binomial expansion of (c - 2) ^ 15 is equal to - 860 , 160 c^2.
To find the 14th term of the binomial expansion of (c - 2)^15, we can use the formula for the term of a binomial expansion: Term(n) = (nCr) * (a^(n-r)) * (b^r). Where n is the exponent of the binomial, r is the term number (starting from 0), nCr is the binomial coefficient, a is the first term of the binomial (c in this case), and b is the second term of the binomial (-2 in this case).
For the 14th term (r = 13), we have: Term(14) = (15C13) * (c^(15-13)) * (-2^13). Using the binomial coefficient formula: 15C13 = (15!)/[(13!)(15-13)!] = 15!/(13! * 2!). Simplifying: 15C13 = (15 * 14 * 13!) / (13! * 2). The (13!) terms cancel out: 15C13 = (15 * 14) / 2 = 105. Substituting the values into the term formula: Term(14) = 105 * (c^2) * (-2^13). Simplifying further: Term(14) = 105 * c^2 * (-8192). Term(14) = -860,160c^2
Therefore, the 14th term of the binomial expansion of (c - 2)^15 is -860,160c^2.
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what is 362,000 rounded to the nearest ten thousand
Answers
Answer:
360,000
Step-by-step explanation:
find the mean of the following,-6,-10,-4,8,9,-2,5,6.
Answers
\(mean = \frac{sum \: of \: all \: observations}{number \: of \: observations} \\ = > mean = \frac{( - 6) + ( - 10) + ( - 4) + 8 + 9 + ( - 2) + 5 + 6}{8} \\ = > mean = \frac{6}{8} \\ = > mean = 0.75\)
This is the answer.
a) Work out (8 × 10³) × (2 x 10¹)
Give your answer in standard form.
Answers
Answer:
8000 times 2x times 1
Step-by-step explanation:
(8 times 10^3) (2xtimes 10^1)
8 times 10^3=8000
= 8000 times 2x times 1
= 160000x
explanation:
(8 times 10^3) (2xtimes 10^1)
8 times 10^3=8000
= 8000 times 2x times 1
= 160000x
what do you call the fixed line of a parabola?
Answers
Answer: directrix.
hoped this helped lmk if it did
Fathi has
$
1.10
$1.10dollar sign, 1, point, 10 in his printing account. Each sheet of paper he uses reduces his printing account balance by
$
0.25
$0.25dollar sign, 0, point, 25. Fathi wants to print out a PDF document that is
47
4747 pages long. To save paper, he decides to print on both sides of each sheet and to print two pages on each side of the sheet.
After Fathi prints, what will be the balance in his printing account?
Answers
Answer:
he can buy 4 sheets of paper to print 16 pages
he will have 0.10 cents left
Step-by-step explanation:
With each piece of paper, he can print 4 pages
He has a full dollar
1/0.25 = 4
he can print 4 pages because the last 10 cents will not be enough to print another page
4 * 4 = 16
He will not be able to print all the pages
Picture attached below, and it's not a rectangle. It's a pentagonsolve a and b
Answers
a)
The perimeter is given by
\(2x+3+4x+5+3x+15+40\)b)
Then we simplify by summing similar terms we obtain that the expression of the perimeter is
\((2x+4x+3x)+(3+5+15+40)\)\(9x+63\)Three angles of a quadrilateral are 650
, 1000, 850
. Find the measure of
fourth angle
Answers
SUM OF THE ANGLES OF A QUADRILATERAL IS 360°
100°+85°+65°=250°
360°=250°+110°
write an equation for the line that gose through the points (-2,6) and (2, 12)
Answers
Answer:
y = \(\frac{3}{2}\) x + 9
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Calculate m using the slope formula
m = \(\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\)
with (x₁, y₁ ) = (- 2, 6 ) and (x₂, y₂ ) = (2, 12 )
m = \(\frac{12-6}{2-(-2)}\) = \(\frac{6}{2+2}\) = \(\frac{6}{4}\) = \(\frac{3}{2}\) , then
y = \(\frac{3}{2}\) x + c ← is the partial equation
To find c substitute either of the 2 points into the partial equation.
Using (2, 12 ) , then
12 = 3 + c ⇒ c = 12 - 3 = 9
y = \(\frac{3}{2}\) x + 9 ← equation of line
Plzz what the answer I’m stuck
Answers
Answer:
the answer to your question is 576 square feet
Step-by-step explanation:
Step 1: 96 ÷ 4 = 24
Step 2: 24×24 =576
The test lasts for 1 hour and 35 minutes. If they finish the test at 1:20, what time did they start the test
Answers
Answer:
15 minutes earlier they started
Mr. Smith is making $25 and hour before his boss gives him a 10% raise. How much will his new hourly wage be?
$25.10
$27.50
$50.00
$26.50
Answers
Answer:
DVvddhsajmsmzmzmxmxmx
Step-by-step explanation:
can someone help please?
Answers
Answer:
Area? or.. perimeter?
Step-by-step explanation:
Separate the funky shape into a rectangle with 2 triangles on each side. Total height of the shape is 18cm, so the height of the triangle is 12 cm (subtract the 6cm bit). Top length and bottom length has a difference of 10cm, so each triangle has a bottom of 5 cm. Using a2+b2=c2, 144+25= 169, or 13. So perimeter is 10+6+6+13+13+20=68cm
If area, then 10x18=180+ ?
(triangle=1/2 x base x height)
=1/2 x 12 x 5= 30
180+60=240
The triangles below lie on the same line. Find the horizontal distance of the smaller triangle. A The horizontal distance of the smaller triangle is 24. B The vertical distance of the smaller triangle is 2. C The horizontal distance of the smaller triangle is 16 . D The vertical distance of the smaller triangle is 16 .
Answers
Answer:
72/100/=18+243*456*43491/45465+451365-42565=
Help me pls I need help with this pleeeeeewwwwwwaaaaaaaaaaase
Answers
Answer:
I am SO sorry I don't have any time to do this for you but here is how
Step-by-step explanation:
Try all of them and see which one is correct
That's literally how you do it.
I just don't have time to do that for you at the moment
Hope this helps :)
(a) Attendance at the Accra Sports Stadium was alysed by the General Secretary, Prosper Harrison Addo. The analysis demonstrated that spectators consisted of 70% males. If seven people are randomly selected from the spectators during a football match, What is the probability that 4 of them are males? (3 marks) i 11. Find the probability that at most 5 of them are females (4 marks)
Answers
a) The probability of randomly selecting 4 males out of 7 spectators, given that 70% of the spectators are males, can be calculated using the binomial probability formula.
b) To find the probability that at most 5 of the randomly selected spectators are females, we need to calculate the cumulative probability of selecting 0, 1, 2, 3, 4, and 5 females from the total number of selected spectators.
a) To calculate the probability of selecting 4 males out of 7 spectators, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- n is the total number of trials (number of people selected)
- k is the number of successful trials (number of males selected)
- p is the probability of success in a single trial (probability of selecting a male)
- C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k! * (n - k)!)
In this case, n = 7, k = 4, and p = 0.70 (probability of selecting a male). Therefore, the probability of selecting 4 males out of 7 spectators is:
P(X = 4) = C(7, 4) * (0.70)^4 * (1 - 0.70)^(7 - 4)
b) To find the probability that at most 5 of the selected spectators are females, we need to calculate the cumulative probability of selecting 0, 1, 2, 3, 4, and 5 females. This can be done by summing the individual probabilities for each case.
P(X ≤ 5 females) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
To calculate each individual probability, we use the same binomial probability formula as in part a), with p = 0.30 (probability of selecting a female).
Finally, we sum up the probabilities for each case to find the probability that at most 5 of the selected spectators are females.
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In November, the rain in a certain valley tends to fall in storms of several days' duration. The unconditional probability of rain on any given day of the month is 0.500. But the probability of rain on a day that follows a rainy day is 0.900, and the probability of rain on a day following a nonrainy day is 0.200. Find the probability of rain on November 1 and 2, but not on November 3.
Answers
If it rained on November 1, then the probability of rain on November 2 is 0.900.
If it did not rain on November 1, then the probability of rain on November 2 is 0.200.
So, we have two cases:
Case 1: Rain on November 1
P(Rain on Nov. 1) * P(Rain on Nov. 2 | Rain on Nov. 1) * P(No Rain on Nov. 3 | Rain on Nov. 2)
= 0.500 * 0.900 * 0.500 = 0.225
Case 2: No Rain on November 1
P(No Rain on Nov. 1) * P(Rain on Nov. 2 | No Rain on Nov. 1) * P(No Rain on Nov. 3 | Rain on Nov. 2)
= 0.500 * 0.200 * 0.900 = 0.090
The probability of rain on November 1 and 2, but not on November 3, is the sum of the probabilities from the two cases:
0.225 + 0.090 = 0.315
Therefore, the probability of rain on November 1 and 2, but not on November 3, is 0.315.
Express the location of the point on the number line as both a fraction and a decimal. A number line that has moving left to right, a starting mark at 0, nine unlabeled tick marks, a tick mark labeled one-tenth, nine unlabeled tick marks, a tick mark labeled two-tenths, nine unlabeled tick marks, and an ending tick mark at three-tenths. There is a dot at the ninth tick mark to the right of one-tenth.
Answers
Answer:
9 / 100 ; 0.09
Step-by-step explanation:
From the line plot given :
The number of unlabeled ticks marks between any two labeled tick marks can be obtained by :
(Difference between any two successive labeled tick marks / number of tick marks)
Picking the labeled tick marks, 0 and 1/10
(1/10 - 0) / 10
1 / 10 ÷ 10
1/ 10 * 1 / 10 = 1 / 100
To find the marked point : (this is tick mark 9 from 0) ;
Number of ticks * distance between ticks
Location of marked tick = 1 / 100 * 9 = 9 / 100
Decimal equivalent = 9/100 = 0.09
For the figure below, give the following. (a) one pair of vertical angles (b) one pair of angles that form a linear pair (c) one pair of angles that are supplementary
Answers
Answer:
Step-by-step explanation:
(a). Opposite angles formed when the two lines cross each other at a point are called as vertical angles.
Therefore, ∠2 and ∠3 is a pair of vertical angles.
(b). Two adjacent angles on a straight line are linear pair of angles.
∠2 and ∠4 are linear pair of angles.
(c). Linear pair of angles are supplementary.
Therefore, ∠2 and ∠4 are supplementary angles.
m∠2 + m∠4 = 180°
Width-16 inches
Length-10 inches
Height- 2 inches
Describe the shape of the cross section when the box is cut parallel to the base.
What is the surface area of the box?
What is the surface area of the box if it is scaled up by a factor of 10?
What is the volume of the box?
What is the volume of the box if it is scaled down by a factor of 1/10?
Answers
Answer:
rectangle
424 in.²
42,400 in.²
320 in.³
0.32 in.³
Step-by-step explanation:
The box has the shape of a rectangular prism.
The cross section of the box is a rectangle.
The length and width are the length and width of the base of the prism.
surface area = perimeter of base × height + 2 × length × width
surface area = 2(length + width) × height + 2 × length × width
surface area = 2(10 in. + 16 in.) × 2 in. + 2 × 10 in. × 16 in.
surface area = 104 in.² + 320 in.²
surface area = 424 in.²
When you scale a solid by a factor of k on a linear measurement, the area is scaled by a factor of k². Since the linear dimensions are scaled by a factor of 10, then the surface area is scaled by a factor os 10² = 100. The surface area of the box scaled by a factor of 10 is 424 in.² × 100 = 42,400 in.²
volume = length × width × height
volume = 10 in. × 16 in. × 2 in.
volume = 320 in.³
When you scale the linear dimensions of a solid by a factor of k, the volume is scaled by a factor of k³. The linear scale factor is 1/10. The change in volume is a factor of (1/10)³ = 1/1000. The original volume is 320 in.³. The scaled volume is 320 in.³ × 1/1000 = 0.32 in.³.
compute the flux of the vector field f = xy, 3yz, 2zx through the portion of the plane 3x 2y z = 6 in the first octant with the downward orientation
Answers
The flux of the vector field F through the portion of the plane 3x + 2y + z = 6 in the first octant with the downward orientation cannot be determined due to the lack of a finite area on the given portion of the plane in the x-y plane.
To compute the flux of the vector field F = (xy, 3yz, 2zx) through the portion of the plane 3x + 2y + z = 6 in the first octant with the downward orientation, we need to evaluate the surface integral of the vector field over the given portion of the plane.
First, we need to parameterize the surface that lies on the plane. Since the equation of the plane is given as 3x + 2y + z = 6, we can express z in terms of x and y as z = 6 - 3x - 2y.
Next, we need to determine the bounds for the variables x and y. Since we are considering the portion of the plane in the first octant, we need to find the values of x and y that satisfy the conditions x ≥ 0, y ≥ 0, and 3x + 2y + z ≤ 6.
Substituting the expression for z in the inequality, we have 3x + 2y + 6 - 3x - 2y ≤ 6, which simplifies to 0 ≤ 0. This condition is always satisfied and does not provide any useful bounds.
Therefore, the portion of the plane in the first octant does not have any boundaries in the x-y plane that define a finite area. As a result, the surface integral and, consequently, the flux of the vector field through this portion of the plane cannot be calculated.
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Please help me fast!!
Answers
Answer:
y=1x+200
Step-by-step explanation:
check by differentiation that y = 3cos3t 4sin3t is a solution
Answers
To check if y = 3cos(3t) + 4sin(3t) is a solution by differentiation, we will differentiate y with respect to t and use the chain rule.
y = 3cos(3t) + 4sin(3t)
dy/dt = -9sin(3t) + 12cos(3t)
The differentiation confirms that the given function y = 3cos(3t) + 4sin(3t) is a valid solution, as we were able to compute its derivative with respect to t without encountering any issues.
To check whether y = 3cos3t 4sin3t is a solution, we need to differentiate it with respect to t and see if it satisfies the differential equation.
y = 3cos3t 4sin3t
dy/dt = -9sin3t + 12cos3t
Now, we substitute y and dy/dt into the differential equation:
d^2y/dt^2 + 9y = 0
(d/dt)(dy/dt) + 9y = 0
(-9sin3t + 12cos3t) + 9(3cos3t 4sin3t) = 0
-27sin3t + 36cos3t + 36cos3t + 27sin3t = 0
As we can see, the equation simplifies to 0=0, which means that y = 3cos3t 4sin3t is indeed a solution to the differential equation.
Therefore, we can conclude that y = 3cos3t 4sin3t satisfies the differential equation and is a valid solution.
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find the supplement of the angle (90-x)
Answers
Answer:
the correct answer is 90•
Step-by-step explanation:
90-x =180
ATQ,
x =180-90
= 90•
find f(x) if f(0) = 3 and the tangent line at (x, f(x)) has slope 3x.
Answers
The answer of the given question based on the differential function is f(x) = (3/2) x² + 3.
Let f(x) be a differentiable function that passes through the point (0,3) and has a tangent line with slope 3x at (x, f(x)).
We know that the tangent line at (x, f(x)) is given by the derivative of f(x) at x, which is denoted by f'(x).
The slope of the tangent line at (x, f(x)) is 3x, which is given as f'(x) = 3x ,
Therefore, we can obtain the function f(x) by integrating f'(x).f'(x) = 3x ,
Integrating both sides with respect to x, we get:
f(x) = (3/2) x² + C, where C is an arbitrary constant.
Using the condition that f(0) = 3, we have:
f(0) = C = 3 ,
Therefore, the function f(x) is:
f(x) = (3/2) x² + 3.
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find the probablility that the sum of two randomly chosen numbers from the set{0,1,2,3,4,5,6,7,8,9} is no grreater than 10
Answers
Answer:
1 /5
Step-by-step explanation:
formulae -
possible outcomes / total chances
2/10
1/5
Complete the square to re-write the quadratic function in vertex form:
y = x2 – 7x + 1
Answers
Answer:
In the vertex form : (x-7/2)²-45
If QT is the perpendicular bisector of PR, find the measure of each.
Answers
Answer:
x = 7
y = 12
PQ = QR = 29
Step-by-step explanation:
In ΔPSR , PS = SR (given)
\(=> 4x+4=7x-17\\=> 7x-4x=17+4\\=>3x=21\\=>x=\frac{21}{3}=7\)
In ΔPQR , QT is perpendicular bisector of PR. So , PT = RT . As ΔPQR has a perpendicular bisector , it is an isosceles triangle.
So,
,\(PQ = RQ\\=>5y-31=2y+5\\=>5y-2y=31+5\\=>3y=36\\=>y=\frac{36}{3} =12\)
Applying the Pythagorean Theorem and the definition of perpendicular bisector, the values of x and y, as well as the measures of each segment in the image are:
x = 7y = 12 QT = 22.7PQ = 29PT = TR = 18QR= 29PS = 32SR = 32Given:
PQ = 5y - 31
PT = 6x - 2y
QR = 2y + 5
PS = 4x + 4
SR = 7x - 17
Find the value of x:
PS = SR (congruent sides)
Substitute\(4x + 4 = 7x - 17\)
Collect like terms and solve for x\(4x -7x = -4 - 17\\\\-3x = -21\)
Divide both sides by -3x = 7
Find the value of y:
PQ = RQ (since QT is the perpendicular bisector of PR, PT equals TR, therefore, triangles PQT and RQT are congruent).
Substitute\(5y - 31 = 2y + 5\)
Collect like terms and solve for y\(5y - 2y = 31 + 5\\\\3y = 36\\\\\mathbf{y = 12}\)
Find the length of each side by plugging in the value of x and y:
PQ = 5y - 31 = 5(12) - 31 = 29
PT = TR = 6x - 2y = 6(7) - 2(12) = 18
QR = 2y + 5 = 2(12) + 5 = 29
PS = 4x + 4 = 4(7) + 4 = 32
SR = 7x - 17 = 7(7) - 17 = 32
Find QT using Pythagorean Theorem:
\(QT = \sqrt{QR^2 - TR^2} \\\\QT = \sqrt{29^2 - 18^2} \\\\QT = 22.7\)
Therefore, applying the Pythagorean Theorem and the definition of perpendicular bisector, the values of x and y, as well as the measures of each segment in the image are:
x = 7y = 12 QT = 22.7PQ = 29PT = TR = 18QR= 29PS = 32SR = 32Learn more here:
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Area pls help its 140 pts
Answers
Answer:
The answer is 126mm
~Please let me know if this is the correct answer! :-) If not, I'm sorry.~
Step-by-step explanation: