(a^3 / 4)^2


Answer 1
Answer: Well, you can change this to (a^3/4)(a^3/4), which is equal to a^6/16, because a^3 times a^3=a^6, and 4 times 4=16.
I hope this helps!
Answer 2
Answer: Multiply the exponents in (a^3/4)^2 =a^3/2 simplified 

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Carl is balancing his checking account. After comparing the bank statement to his register, he notices an outstanding debit of $58.00.Which shows the correct amount in Carl’s checking account?$51.75$109.75$167.75$221.87
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Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges $75 a week plus $0.25 a mile. How many miles can they travel and still keep within their $200 budget?


Answer: 500 miles

Step-by-step explanation:

Given : Sergio and Lizeth have planned to rent a car from a company that charges $75 a week plus $0.25 a mile.

i.e. Fixed charge= $75

Rate per mile = $0.25

Let x denotes the number of miles.

Then, Total charges = Fixed charge+ Rate per mile x No. of miles traveled

=  $75+ $0.25x

To keep budget within $200, we have following equation.

75+0.25x=200\n\n\Rightarrow\ 0.25=200-75\n\n\Rightarrow\ 0.25=125\n\n\Rightarrow\ x=(125)/(0.25)=(12500)/(25)=500

Hence, they can travel 500 miles and still keep within their $200 budget.

Mang Jose plans to fence his rectangular lot before he will plant mushrooms for his mushrooms production business. The perimeter of the lot is 40 meters and the area is 96 square meters.Using the concept of the sum and product of roots of a quadratic equation, how would you determine the length and the width of the rectangular lot? Provide a quadratic equation representing this scenario.

Someone help me


Answer: The rectangular lot is 12x8 meters

Step-by-step explanation:Perimeter of a geometric figure is the sum of all its sides.

A rectangle is a quadrilateral that has opposite sides parallel and equal, which means, and suppose l is length and w is width:

P = 2l + 2w

The perimeter of the lot is 40m, thus:

2l + 2w = 40

Area of a rectangle is calculated as:

A = length x width

The lot has area of 96, thus:

lw = 96

Solving the system of equations:

2l + 2w = 40 (1)

lw = 96 (2)

Isolate l from (1):

2l = 40 - 2w

l = 20 - w (3)

Substitute (3) in (2):

w(20-w) = 96



There are many methods to determine the roots of a quadratic equation. One of them is using the sum and product of those roots.

  • Sum of the roots is given by:

sum = (-b)/(a)


sum = 20

  • Product of the roots is:



prod = 96

The roots of the quadratic equation are numbers which the sum results in 20 and product is 96:

w₁ = 12

w₂ = 8

If we substitute w to find l, the numbers will be l₁ = 8 and l₂ = 12.

Since length is bigger than width, the rectangular lot Mang Jose has to plant mushrooms measures 12m in length and 8m  in width

The top of a molehill is 4in above ground level.The bottom of a mole’s burrow is -9in relative to ground level.
What is the distance between the top of the molehill and the burrow? Show your work.



  • 13 inches

Step-by-step explanation:

The distance is the difference in top and bottom positions:

  • 4 - (-9) = 4 + 9 = 13 inches

Use the inner product〈f,g〉=∫10f(x)g(x)dxin the vector space C0[0,1] of continuous functions on the domain [0,1] to find 〈f,g〉, ∥f∥, ∥g∥, and the angle αf,g between f(x) and g(x) forf(x)=−10x2−6 and g(x)=−9x−4.〈f,g〉= ,∥f∥= ,∥g∥= ,αf,g .



a) <f,g> = 2605/3

b) ∥f∥ = 960

c) ∥g∥ = 790

d) α = 90  


a) We calculate  <f,g> using the definition of the inner product:

<f,g> = \int\limits^1_0 {10(-10x^(2) -6)(-9x-4)} \, dx \n        \n        =\int\limits^1_0 {900x^(3)+400x^(2) +540x+240 } \, dx\n    \n      = (225x^(4) + (400x^(3) )/(3) + 270x^(2)   +240x)\n      = (2605)/(3)

b) How

∥f∥ = <f,f> then:

∥f∥ = <f,f> = \int\limits^1_0 {10(-10x^(2) -6)(-10x^(2) -6)} \, dx \n        \n        =\int\limits^1_0 {1000x^(4)+1200x^(2) + 360} \, dx\n    \n      = (200x^(5) + 400x^(3) +  360x)\n      = 960


∥g∥ = <g,g>

∥g∥ = <g,g> = \int\limits^1_0 {10(-9x-4)(-9x-4)} \, dx \n        \n        =\int\limits^1_0 {810x^(2)+720x + 160} \, dx\n    \n      = (270x^(3) + 360x^(2) +  160x)\n      = 790

d) Angle between f and g

<f,g> = ∥f∥∥g∥cosα


\alpha = cos^(-1)((2605/3)/((790)(960)) )\n\n\alpha = 90

Final answer:

The answer to this problem involves applying integrals, norms, and concepts of angles between vectors to the functions f(x) and g(x). The INNER PRODUCT is the integral of the products of the two functions, the norms are the square roots of the inner products of the functions with themselves, and the angle between the functions is calculated using the dot product and norms.


To find the inner product 〈f,g〉, the norms ∥f∥ and ∥g∥, and the angle αf,g between the functions f(x)=−10x2−6 and g(x)=−9x−4, we'll apply concepts from vector calculus. The inner product (also known as the dot product) is the integral from 0 to 1 of the products of the two functions. The norm of a function is the square root of the inner product of the function with itself. The angle between two vectors in a Vector Space, in this case the space of continuous functions C0[0,1], is given by cos(α) = 〈f,g〉/( ∥f∥∙ ∥g∥). Integrating and solving these equations will give us the desired values.

Learn more about Vector Calculus here:



Please help Dekalbs current population is 3,500 and it is increasing by 40 people pr year. Sycamores current population is 12,040 and it is decreasing by 100 people every year

Q: Write expressions for each population if x represents the number of years
And What is the current population of dekalb?



Dekalb: 3500+40x = F

Sycamores : 12,040 - 100x = F

Step-by-step explanation:

These expressions can be used to explain each population

X = number of years for both expression

F = Final population

Dekalb: 3500+40x = F

Sycamores : 12,040 - 100x = F

The current population if 3500 if no years have gone by, plug in the year of numbers for X to find final population after the amount of years.

Lena is on a two week bicycle trip. After 5 days she had ridden 212 miles. Express Lena’s rate as a unit rate



42.4 miles per day

Step-by-step explanation:

212 miles ÷ 5 (days) = 42.4 miles

which means Lena can go at the rate of

42.4 miles per day