Answers
The inverse of the function f(x) = √(x+1) is [tex]f^-1[/tex](x) = x² - 1.
What is inverse function?Given a function f(x), its inverse function, denoted as [tex]f^-1[/tex](x), is a function that takes the output of f(x) and returns the original input. In other words, if y = f(x), then x = [tex]f^-1[/tex](y).
According to question:To find the inverse of the function f(x), we follow these steps:
Step 1: Replace f(x) with y:
y = √(x+1)
Step 2: Interchange x and y:
x = √(y+1)
Step 3: Solve for y:
x² = y+1
y = x² - 1
Step 4: Replace y with f^-1(x):
f^-1(x) = x² - 1
Therefore, the inverse of the function f(x) = √(x+1) is [tex]f^-1[/tex](x) = x² - 1.
To verify this, we can check that ([tex]f^-1[/tex] o f)(x) = x and (f o [tex]f^-1[/tex])(x) = x for all x in the domain of the functions:
([tex]f^-1[/tex] o f)(x) = [tex]f^-1[/tex](f(x)) = (√(x+1))² - 1 = x
(f o [tex]f^-1[/tex])(x) = f([tex]f^-1[/tex](x)) = √((x² - 1) + 1) = √(x²) = |x| + 1
Since the range of f(x) is [0, infinity), we restrict the domain of [tex]f^-1[/tex](x) to [1, infinity) to ensure that the inverse is a function.
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Related Questions
HELP ASAP ! Given the rectangle in this example, and a scale factor of 3.5, what are the side lengths of the new rectangle?
Answers
Answer:
Length: 22 inches × 3.5 = 77 inches
Width: 10 inches × 3.5 = 35 inches
Devonte is studying for a history test he uses 1/8 of a side of one sheet of paper to write notes for each history event he fills 2 full sides of one sheet paper. which expression could be used to find how many events
Answers
The expression to find the number of events is: (1 event) / (1/8 side) = (n events) / (1/4 sides).
Devonte is studying for a history test and uses 1/8 of a side of one sheet of paper to write notes for each history event. He fills 2 full sides of one sheet of paper. To find out how many events he wrote notes for, you can set up an expression using the given information.
Since Devonte uses 1/8 of a side for each event, and he fills 2 sides, you can calculate the total amount of space he used by multiplying the fractions: (1/8) * 2. This simplifies to 2/8 or 1/4. Now, you can set up a proportion to find the number of events (n) that Devonte wrote notes for:
(1 event) / (1/8 side) = (n events) / (1/4 sides)
Cross-multiply to solve for n:
1 * (1/4) = n * (1/8)
1/4 = n/8
To find n, multiply both sides by 8:
(8) * (1/4) = n
2 = n
So, Devonte wrote notes for 2 history events using the 2 full sides of one sheet of paper. The expression to find the number of events is: (1 event) / (1/8 side) = (n events) / (1/4 sides).
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Complete Question:
Devonte is studying for a history test. He uses 1/8 of a side of one sheet of paper to write notes for each history event. He fills 2 full sides of one sheet of paper. Which expression could be used to find how many events Devonte makes notes for?
Triangle XYZ undergoes a dilation to produce triangle X'Y'Z'. The coordinates of both triangles are shown.
X(8.2)→ X'(4,1)
Y(-10,4)→Y'(-5,2)
Z(-2,0) Z’(-1,0)
What is the scale factor of the dilation?
Answers
The scale factor used in the dilation of the triangles is 1/2
Determining the scale factor used.From the question, we have the following parameters that can be used in our computation:
XYZ with vertices at X(8, 2)X'Y'Z' with vertices at X'(4, 1)The scale factor is calculated as
Scale factor = X'/X
Substitute the known values in the above equation, so, we have the following representation
Scale factor = (4, 1)/(8, 2)
Evaluate
Scale factor = 1/2
Hence, the scale factor is 1/2
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There are 90 children in year 6 at woodland junior school
they are split into three classes
class
number n class
27
6m
6p
33
6t
30
each child chose football or netball or hockey.
in 6m, 13 children chose hockey.
the rest of the class were split equally between football and netball.
in 6p, 9 children chose netball
twice as many children chose football as chose hockey
in 6t the ratio of children who chose
football to netball to hockey was 1:2:3
complete this table
class
number in class
football
netball
hockey
6м
27
13
6p
33
6t
30
Answers
In Year 6 at Woodland Junior School, there are 90 children split into three classes of 14, 24, and 12 on the basis of there selection of sports. In 6M, 14 not chose hockey, and the rest of the class was split equally between football and netball. In 6P, 24 not chose netball, 16 chose football, and 8 choose hockey In 6T, the ratio of football to netball to hockey was 1:2:3. The completed table is shown.
To complete the table, we need to distribute the remaining children who did not choose their sport in each class. Here's how we can calculate it
In 6M, the number of children who did not choose hockey is 27 - 13 = 14.
Since the rest of the class was split equally between football and netball, each of these two sports will have 14/2 = 7 children.
In 6P, the number of children who did not choose netball is 33 - 9 = 24.
Since twice as many children chose football as chose hockey, we can write the number of footballers as 2x, and the number of hockey players as x. Then we have 2x + x + 9 = 33, which gives x = 8. Therefore, we have 16 children who chose football, and 8 children who chose hockey. The number of children who did not choose any of these two sports is 33 - 16 - 8 - 9 = 0.
In 6T, the ratio of children who chose football to netball to hockey was 1:2:3. Let's call the number of children who chose netball as 2x, and the number of children who chose hockey as 3x. Then we have x + 2x + 3x = 30, which gives x = 6. Therefore, we have 6 children who chose football, 12 children who chose netball, and 18 children who chose hockey.
Thus, the completed table is shown.
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If x = -3, then which inequality is true?
Answers
The inequailty y < x + 3 would be true when x = -3 and all values of y are less than 0
If x = -3, then which inequality is true?From the question, we have the following parameters that can be used in our computation:
The statement that x = -3
The above value implies that we substitute -3 for x in an inequality and solve for the variable y
Take for instance, we have
y < x + 3
Substitute the known values in the above equation, so, we have the following representation
y < -3 + 3
Evaluate
y < 0
This means that the inequailty y < x + 3 would be true when x = -3 and all values of y are less than 0
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What is the volume of the prism, measured in cubic inches
Answers
Answer:
360 cubes
Step-by-step explanation:
1 ml =
a
litres
ii)
b
ml = 1 litre
iii) 1 cl =
c
litres
iv)
d
cl = 1 litre
v) 1 cl =
e
ml
vi)
f
cl = 1 ml
Answers
The corresponding measure of the parameters are;
i. 1ml = 0. 001 liter a.
ii. 1000ml = 1 liter b.
iii. 1 cl = 0. 01 liter c.
iv. 10dcl = 1 liter d.
v. 1cl = 100ml e.
v. 0. 01 cl = 1ml f.
How to determine the valuesTo convert the factors, we need to know the following conversion rates.
We have;
1 milliliter = 0. 001 liter
1 centiliter = 0. 01 liter
1 deciliter = 0. 1 liter
1 cubic centimeter = 1 millimeter
Hence, the sizes are determined by the corresponding factor.
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Complete question:
Convert the following to their equivalent measurement for each letter
i. 1 ml = a liters
ii) b ml = 1 liters
iii) 1 cl = c liters
iv)d cl = 1 liters
v) 1 cl = e ml
vi) f cl = 1 ml
Mr. jackers was transporting a giraffe from one zoo to another. what is a reasonable amount of the weight of his cargo?
a. 200 lbs.
b. 400 lbs.
c. 1 ton
d. 2 tons
Answers
The reasonable amount of weight of his cargo Mr. Jackers while transporting a giraffe from one zoo to another is 1 ton and 2 tons. Options C and D are correct.
To arrive at a reasonable estimate, we can consider the average weight of a giraffe, which ranges between 2,600 lbs and 4,250 lbs for adult males and females, respectively.
Therefore, a reasonable amount for the weight of Mr. Jackers' cargo would be between 2,600 lbs and 4,250 lbs, which translates to either 1 ton or 2 tons.
Thus, options c and d are the only reasonable answers, with d (2 tons) being the safer choice.
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Which of the following equations are equivalent? Select three options. 2 + x = 5 x + 1 = 4 9 + x = 6 x + (negative 4) = 7 Negative 5 + x = negative 2
Answers
Answer:
The three equivalent equations are x+1=4, -5+x=-2, and 2+x=5. The x equals 3.
Step-by-step explanation:
Over what interval is the function shown in the table increasing? Decreasing? Y=6x2
Answers
Decreasing over the x-interval of 0(less than or equal to sign)x(less than it equal to sign)2
The carpet for the office you completed cost $576. If you're using the same carpet in the new client's bedroom, how much will it cost? Round up to the nearest whole dollar. (Hint: You may want to find the area of each room and the cost of the carpet per square foot. )
Answers
The cost of a similar carpet in the client's bedroom is $225 when the area of the bedroom is 120 square feet.
Given Data:
Cost of Carpet = $576
length of office room = 16
Width of office room = 19.2
length of Bedroom = 12
Width of Bedroom = 10
We need to determine the area of the office and the bedroom. we can determine it by using the Area of a rectangle formula which is given as,
Area of a rectangle = length × width
Area of Office = length of office room × width of office room
= 16 × 19.2
= 307.2 square feet
Area of Bedroom = length of Bedroom × width of Bedroom
= 12 × 10
= 120 square feet
Therefore, the area of the office is 307.2 square feet, and the area of the Bedroom is 120 square feet.
From the above data, the area of the office is 307.2 square feet and the carpet will cost $576 for the office. we need to find the cost of the carpet for the 120 square feet bedroom, let's assume the cost as x, and cross multiply we get,
x = 120 × 576 / 307.2
x = 69120 / 307.2
x = $225
Therefore, the cost of the bedroom carpet is $225.
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E
josh needs to find the measures of the
angles on a barn's trusses.
find the measure of abc if abd is a right angle and cbd is 60 degrees. write and solve a subtraction equation
Answers
The measure of angle ABC is 30 degrees.
How to find the measure of angle ABC?To find the measure of angle ABC, we need to use the fact that the sum of angles in a triangle is always 180 degrees. Since we know that angle ABD is a right angle (90 degrees) and angle CBD is 60 degrees, we can find the measure of angle ABC as follows:
angle ABD + angle CBD + angle ABC = 180 degrees
Substituting the given values, we get:
90 degrees + 60 degrees + angle ABC = 180 degrees
Simplifying, we get:
150 degrees + angle ABC = 180 degrees
Now we can solve for angle ABC by subtracting 150 degrees from both sides:
angle ABC = 180 degrees - 150 degrees
angle ABC = 30 degrees
Therefore, angle ABC measures 30 degrees.
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The volumes of two similar solids are 857.5 mm^3 and 540 mm^3. The surface area of the smaller solid is 108 mm^2. What is the surface area of the larger solid?
*
147 mm^2
68 mm^2
16 mm^2
216 mm^2
Answers
Answer:
147 mm^2
Step-by-step explanation: The surface area has to be more, but not double the smaller figures surface area therefore the answer is 147 mm^2
The requried surface area of the larger solid is approximately 147 mm². Option A is correct.
What is surface area?The surface area of any shape is the area of the shape that is faced or the Surface area is the amount of area covering the exterior of a 3D shape.
Let's call the larger solid's surface area S.
Since the two solids are similar, their volumes have a ratio of (side length)³. Let's call the ratio of the side lengths of the larger to the smaller solid as k. Then:
[tex](k^3)(540 mm^3) = 857.5 mm^3[/tex]
Simplifying the above equation, we get:
[tex]k = (857.5/540)^{(1/3)}[/tex] =7/6
So, the larger solid is about 7/6=1.183 times bigger than the smaller solid in terms of side length. Since the surface area has a ratio of [tex](side length)^2[/tex], we can find the surface area of the larger solid by:
[tex]S = (1.183^2)(108 mm^2) \approx 147 mm^2[/tex]
Therefore, the surface area of the larger solid is approximately 147 mm². Answer: A.
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Complete the description of a real-world situation that might involve three linear equations in three variables.
you are trying to find the ages of three people. you know the sum of all three ages, the sum of the first two ages and blank (the answer choices are twice the third or the third age squared), and the sum of the first and third ages and blank (the answer choices are twice the second or the square root of the second)
* just to be clear there are two blanks and two possible answer choices for each
Answers
A real-world situation that might involve three linear equations in three variables is trying to determine the ages of three siblings.
Let's call them A, B, and C. We know that the sum of all three ages is a certain value, let's say it's 60. We also know the sum of the first two ages is either twice the third age or the third age squared. For example, if the sum of the first two ages is twice the third age, we could write it as A + B = 2C.
Alternatively, if the sum of the first two ages is the third age squared, we could write it as A + B = C^2. Similarly, we know the sum of the first and third ages is either twice the second age or the square root of the second age. So, we could write it as A + C = 2B or A + C = sqrt(B).
We now have three linear equations in three variables that we can use to solve for the ages of the three siblings. By solving the system of equations, we can find out how old each sibling is.
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A snail moves about 0.013m each second. About how many hours would it take the snail to travel 174km?
Answers
Answer:
Step-by-step explanation:
0. 587. Write this value in
scientific notation.
Answers
. 0.587 in scientific notation is 5.87 x 10^-1.
Find out the scientific notation of the given value?Scientific notation is a way of writing numbers that are very large or very small in a compact and standardized way. In scientific notation, a number is expressed as a coefficient multiplied by 10 raised to some power.
For example, the number 0.587 can be written in scientific notation as 5.87 x 10^-1. The coefficient 5.87 is obtained by moving the decimal point one place to the right, while the negative exponent -1 indicates that the decimal point has been moved one place to the left, to the tenth place.
Scientific notation is commonly used in scientific and engineering fields where very large or very small numbers are often encountered, and it allows for easier calculation and comparison of these values.
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A styrofoam model of a volcano is in the shape of a cone. The model has a circular base with a diameter of 48 centimeters and a height of 12 centimeters. Find the volume of foam in the model to the nearest tenth. Use 3. 14 for TT.
Answers
The volume of foam in the model is approximately 27,211.5 cubic centimeters
The radius of the circular base can be found by dividing the diameter by 2:
radius = diameter / 2 = 48 cm / 2 = 24 cm
The formula for the volume of a cone is:
V = (1/3) * π * r² * h
where π is approximately 3.14, r is the radius of the circular base, and h is the height of the cone.
Substituting the values we have:
V = (1/3) * 3.14 * 24² * 12
V = 27,211.52 cm³
Rounding this to the nearest tenth, we get:
V ≈ 27,211.5 cm³
Therefore, the volume of foam in the model is approximately 27,211.5 cubic centimeters
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(12.7)
2. A swimming pool is in the shape of a rectangular
prism with a horizontal cross-section 10 feet by 20
feet. The pool is 5 feet deep and filled to capacity.
Water has a density of approximately 60 pounds
per cubic foot
What is the approximate mass of water in the pool?
A. 8,000 lb.
B.
12,500 lb.
C
16,700 lb.
D. 60,000 lb.
Answers
Answer:
Step-by-step explanation:
The volume of the pool can be calculated as:
Volume = length x width x height
Volume = 10 ft x 20 ft x 5 ft
Volume = 1000 cubic feet
The mass of the water in the pool can be calculated as:
Mass = Volume x Density
Mass = 1000 cubic feet x 60 pounds/cubic foot
Mass = 60,000 pounds
Therefore, the approximate mass of water in the pool is 60,000 lb , which corresponds to option D.
Jaime is cutting shapes out of cardboard to make a piñata. One of the shapes is shown in a
coordinate grid
c. (0,10)
d. (3,2)
e. (9,0)
f. (3,-2)
g (0,-10)
h. (3,-2)
a. (-9,0)
(it’s the shape of a star)
What is the length of side AB? Round your answer to the nearest tenth of a unit.
Show your work.
Answers
The length of side AB is 6.3 units.
How to find the length of side ABThe length of side AB is solved using the distance formula below
AB = √((x₂ - x₁)² + (y₂ - y₁)²
where
(x₁, y₁) = (-9, 0) and
(x₂, y₂) = (-3, 2).
AB = √((-3 - (-9))² + (2 - 0)²)
AB = √(6² + 2²)
AB = √(40)
AB = 2√(10)
AB = 6.3245
AB = 6.3 to the nearest tenth
Therefore, the length of side AB is 6.3 units.
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Which shows 71. 38 in word form? O A seventy-one thirty-eighths O B. Seventy-one and thirty eighths O c. Seventy-one and thirty-eight tenths D. Seventy-one and thirty-eight hundredths E seventy-one and thirty-eight thousands
Answers
The number 71.38 can be written in word form as "seventy-one and thirty-eight hundredths." The correct answer is option D.
In decimal notation, the number 71.38 can be broken down into its whole number and decimal parts. The whole number part is 71, and the decimal part is 0.38.
In a decimal number, the digits to the right of the decimal point represent fractions of a whole. Each digit to the right of the decimal point has a place value that is a power of 10.
In word form, the decimal part 0.38 is read as "thirty-eight hundredths." Therefore, when combined with the whole number 71, the correct word form is "Seventy-one and thirty-eight hundredths."
Therefore option D is the correct answer.
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If john buys 8 jackets at x dollars a piece and 2 movie tickets at $11 a piece, whats the most he can spend on each jackets if his total budget is $200
Answers
Answer:
22.25
Step-by-step explanation:
8x+ (11*2)=200
first, multiply 11 and 2 =22
that leaves 8x+22=200
so now let's subtract the 22 from both sides
8x=178 now divide the 8
x= 22.25 so that is the most the jackets could have cost
Redland Municipal Middle School offers the following girls sports:
Part A:
Regina wants to play one sport in each of the three seasons during her 8th grade year. Create a list that shows the sample space for the sports that she could play.
Part B:
What is the probability that Regina plays tennis and/or basketball during her 8th grade year?
Answers
probability that Regina plays tennis and/or basketball during her 8th grade year at Redland Municipal Middle School, we first need to know the total number of sports offered and the number of students participating in each sport. However, based on the information provided in your question, we do not have sufficient data to determine the probability.
In general, to determine the probability, we would follow these steps:
1. Determine the total number of sports offered at the school (let's call this S).
2. Find out the number of students participating in tennis (T) and basketball (B).
3. Calculate the probability of Regina playing tennis (P_T) by dividing T by S.
4. Calculate the probability of Regina playing basketball (P_B) by dividing B by S.
5. Use the formula P(T or B) = P_T + P_B - P(T and B) to calculate the probability that Regina plays tennis and/or basketball. Here, P(T and B) is the probability of her playing both sports.
Without the specific numbers, we cannot provide a precise probability. Please provide more details about the sports and participation at Redland Municipal Middle School to help us calculate the probability accurately.
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Nick used his credit card to make several large purchases, and now owes the credit card company $4,273 plus 32% interest. His mother gave him $500 for h
birthday. Which is the most responsible choice he can make regarding his birthday money?
A He can put $250 into a savings account that pays 3. 8% interest and pay down his credit card debt with the rest.
B. He can put all the birthday money in the 3. 8% savings account
о
C. He can use all the money to pay down his credit card debt.
O D. He can invest $200 in the stock market and pay down his credit card debt with the rest.
Answers
The most responsible choice for Nick regarding his birthday money would be option C, which is to use all the money to pay down his credit card debt.
This is because credit card debt generally comes with high- interest rates, and it can be grueling to pay off the debt if the interest keeps adding up. By using the$ 500 to pay down his credit card debt, Nick can reduce the quantum he owes and, as a result, pay lower interest in the long run. Options A and D suggest unyoking the plutocrat between paying down debt and investing/ saving, which may not be the stylish choice given Nick's current fiscal situation.
Option B suggests putting all the plutocrat into a savings regard with a lower interest rate, which would not be as effective in reducing his credit card debt. thus, option C is the most responsible choice for Nick.
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sum of 3 consecutive even numbers is 18
Answers
Answer:
Step-by-step explanation:
5 6 7
Find the derivative y = cos(sin(14x-13))
Answers
To find the derivative of y = cos(sin(14x-13)), we will use the chain rule.
Let's start by defining two functions:
u = sin(14x-13)
v = cos(u)
We can now apply the chain rule:
dy/dx = dv/du * du/dx
First, let's find dv/du:
dv/du = -sin(u)
Next, let's find du/dx:
du/dx = 14*cos(14x-13)
Now we can put it all together:
dy/dx = dv/du * du/dx
dy/dx = -sin(u) * 14*cos(14x-13)
But we still need to substitute u = sin(14x-13) back in:
dy/dx = -sin(sin(14x-13)) * 14*cos(14x-13)
So the derivative of y = cos(sin(14x-13)) is:
dy/dx = -14*sin(sin(14x-13)) * cos(14x-13)
To find the derivative of the function y = cos(sin(14x - 13)), we can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
Let u = sin(14x - 13), so y = cos(u). Now we find the derivatives:
1. dy/du = -sin(u)
2. du/dx = 14cos(14x - 13)
Now, using the chain rule, we get:
dy/dx = dy/du × du/dx
dy/dx = -sin(u) × 14cos(14x - 13)
Since u = sin(14x - 13), we can substitute back in:
dy/dx = -sin(sin(14x - 13)) × 14cos(14x - 13)
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Your friend purchased a medium pizza for 10. 32 with a 30% off coupon what is the price of the pizza with out a coupon
Answers
The price of the medium pizza without the coupon is approximately $14.74.
To find the original price of the pizza without the coupon, we can use the following formula:
Original Price = Discounted Price / (1 - Discount Percentage)
In this case, the discounted price is $10.32 and the discount percentage is 30% or 0.3. Plugging the values into the formula:
Original Price = $10.32 / (1 - 0.3)
Original Price = $10.32 / 0.7
Original Price ≈ $14.74
So, the price of the medium pizza without the coupon is approximately $14.74.
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Adcb is a rectangle. ac = 16 and bd = 2x + 4, find the value of x.
Answers
In a rectangle, the diagonals are equal in length. So we can write the equation: AC = BD or 16 = 2x + 4. Solving for x, we get x = 6.
1) Find the value of the k that makes the sequence arithmetic
-13,k,-3,2,...
A)5
B)-8
C)-5
D)-10
2) Which formula describes the arithmetic sequence?
-4/5, -3/5,-2/5,-1/5...
A)tn=-n/5
B)tn=n/5-1
C)tn=1+n/5
D)tn=-n/5-1
3)What're the next 4 terms in the series?
t1=3,tn=1/2,tn-1 -1
A)1/2, -3/4, -11/8, -27/10
B)3/2, 3/4, 3/8, 3/16
C)-1/2, 3/4, 5/8, 21/16
D)2, 0, -1, -3/2
4)Which formula describes the arithmetic sequence?
-3,-6,3,-6
A)tn=-tn-1+3
B)tn=-tn-1-1
C)tn=tn-1-1
D)-tn-1-3
Answers
Solutions of Two-Variable Linear Inequalities
-40
5
-5
40
X
Select all of the following ordered pairs that are in
the solution set of the inequality shown.
(-10,-10)
0
0
0
(0.
(0, 10)
(10,0)
(20, 0)
(100, 100)
Answers
Answer: -403
Step-by-step explanation:
simplify
(2a² b c³) (-a² b⁵)
pls help asap!
Answers
Answer:
[tex]-2a^{4}b^{6} c^{3}[/tex]
Step-by-step explanation:
(2a² b c³) (-a² b⁵)
distribute amongst each other
-2a²bc³a²b⁵
rearrange
-2a²a²bb⁵c³
combine like terms
[tex]-2a^{4}b^{6} c^{3}[/tex]
Hope this helps!
Answer: -2[tex]a^{4}[/tex][tex]b^{6}[/tex]c³
Step-by-step explanation:
Given:
(2a² b c³) (-a² b⁵)
Rules:
>Multiply numbers first
>If a number with an exponent is multiplied to another, you can combine if the bases are the same(lower number)
>If they have the same base, you can add the exponents
Solution:
(2a² b c³) (-a² b⁵) >multiply the - and 2
-2(a² b c³) (a² b⁵) > combine the a's by adding the expnents 2+2
-2[tex]a^{4}[/tex]( b c³) ( b⁵) > combine b, there is an invisible 1 by the first b 1+5
-2[tex]a^{4}[/tex][tex]b^{6}[/tex]c³ >c stays as is because nothing to combine with