
Math
Given the expression E = 200m^4 + 80m^3 + 8m^2. We need to simplify as mush as possible. First we will need to factor all terms is order to find a common factor. Let us factor each term. 200m^4 =...

Math
Let the integers be x, y and z. Given that the sum of the three integers is 66. ==> x + y + z = 66..............(1) Also, given that the second integer (y) is 2 more than the first (x). ==> y...

Math
Using the properties of absolute value yields: Hence, evaluating the solutions to absolute value equation yields and .

Math
Given y= (x2) / (x^2  3) We need to find y'. Since y is a quotient, then we will use the quotient rule to find the derivative. ==> Let y= u/v such that: u= x2 ==> u' = 1 v= x^2 3...

Math
You need to evaluate the determinant of the matrix to check if there exists a inverse for the given matrix, such that: `det A = [(1,2,1),(3,7,1),(5,7,15)] = 26 != 0` Since the determinant of...

Math
The right triangle has two sides of length 4 and 3. The third side can have a length which can be calculated using the Pythagorean Theorem. As it is not given that 4 and 3 are the lengths of the...

Math
The sum of 3/4 + 3/20 can be calculated in the following way 3/4 + 3/20 make the denominator the same, here it would be 20 3*5 / 4*5 + 3/ 20 => 15/20 + 3/ 20 => (15 + 3) / 20 => 18/20...

Math
To find which value if greater, we will calculate the value of each percentage. ==> 10% of 325 = 10/100 * 325 = 32.5 ==> 15% of 295 = 15/100 * 295 = 44.25 ==> 25% of 250 = 25/100 * 250 =...

Math
In a decimal number each digit is a power of 10, for example in 123 is equal to 1*10^0 + 2*10^1 + 3*10^2. For binary numbers the base is 2 instead of 10. So 11001011 = 1*2^0 + 1*2^1 + 0*2^2 + 1*2^3...

Math
If an amount P is invested for n terms at a rate of interest of r, it grows to P*(1 + r)^n. Here, the annual rate of interest is 4%, as the compounding is done semiannually, the effective rate for...

Math
The area of a triangle if two sides and the enclosed angle are given is (1/2)*s1*s2*cos A Here the sides are 6 yd and 2 yd and the enclosed angle is 10 degrees. The area of the triangle is...

Math
The equation to be solved is 8^(6x+5)2/8=0 8^(6x+5) = 2/8 => 2^3^(6x + 5) = 1/4 = 2^2 As the base 2 is the same, equate the exponent 3*(6x + 5) = 2 18x + 15 = 2 => 18x = 17 => x =...

Math
Let the width of the rectangle be x, then the length is 3x +2. Now we are given that the area if 33. Then we know that x*(3x+2) = 33 Let us solve. ==> 3x^2 + 2x = 33 ==> 3x^2 +2x  33 = 0...

Math
We know that cos (pi/6) = (sqrt 3)/2 cos (pi/6) = cos 2*(pi/12) = 2*[cos (pi/12)]^2  1 => (sqrt 3)/2 = 2*[cos (pi/12)]^2  1 => 2*[cos (pi/12)]^2 = 1 + (sqrt 3)/2 => [cos (pi/12)]^2 = 1/2...

Math
We have to solve the equation: x/(x+4) + 4/(x+4) + 2 = 0 x/(x+4) + 4/(x+4) + 2 = 0 => x/(x+4) + 4/(x+4) = 2 => (x + 4)/(x + 4) = 2 => 1 = 2 This is not possible. The given equation has...

Math
The endpoints of the diameter of the circle are (4, 7) and (7, 11). The distance between the points or the length of the diameter is given as: sqrt[ (4  7)^2 + (7  11)^2] => sqrt[3^2 + 4^2]...

Math
We have to find the value of Sum(100 + k), for k = 1 to 100 Sum(100 + k), k = 1 to 100 => 100*100 + Sum(k), k = 1 to 100 The sum of the numbers from 1 to n is given by n*(n + 1)/2 => 100*100...

Math
First (A + B + C)/3 = 2709 so (1345 + 568 + x)/3 = 2709 1345 + 568 + x = 3*2709 1913 + x = 8127 x = 8127  1913 x = 6214 Check  (1345 + 568 + 6214)/3 = 8127/3 = 2709 So the answer is 6214.

Math
The answer is if a>b>0 then a^n > b^n Proof by induction a) a>b ==> ````a^1 > b^1 by definiton.b) suppose a^(n1) > b^(n1)c) We know that if a>b>0 and c>d>0 then...

Math
Given the lines: ax + by + c = 0 We need to determine the x and y intercepts and the gradient. The xintercept is when the line meets the xaxis. Then, y= 0 ==> ax + c = 0 ==> x = c/a Then...

Math
1. Simplify `(a^2b^0c)^0` Anything to raised to the zero is equal to one, so the answer here is 1. 2. Simplify `x^{2}y^3` We can write `x^{2}` as `\frac{1}{x^2}` ,so this simplifies to...

Math
Zeller's algorithm is ` h=(q+__((m+1)26)/10__+Y+__Y/4__+__C/4__2C) mod 7` Where q is the date of the month m is the month (March=3, April=4, ... January=13, February=14) Y is the year of the...

Math
`2(XY)^2  9(XY)Z  5Z^2 = 0` Multiply by 1 both sides. `==> 5z^2 +9(xy)z 2(xy)^2 = 0` Let (xy)= A ==> `5z^2 +9Az 2A^2 = 0` `==> (5z +A)(z2A) = 0` `==> (5z+(xy))(z2(xy)) =...

Math
`f(x)=2(x^21)^3` `f'(x)=2*3(x^21)^2(2x)=12x(x^21)^2` `f''(x)=12(x^21)^2+12x(2)(x^21)(2x)` `=12(x^21)^2+48x^2(x^21)` `=60x^472x^2+12` As a check, expand the binomial and then take the...

Math
`a_1 = 3 ` `a_(k+1)=a_k^2` `a_2 = a_1^2 = (3)^2 = 9` `a_3 = a_2^2 = (9)^2 = 81` `a_4 = a_3^2 = (81)^2 = 6561` `a_5 = a_4^2 = (6561)^2 = 43046721` So the answer is 3, 8, 81, 6561, 43046721

Math
`f(x)= x^3 5x^2 +6x +3` We know that the slope of the tangent line is the derivative at the point of tendency. But we know that the tangent line is horizontal. Then, the slope is 0. `==>...

Math
The vertex of the quadratic function f(x)=ax^2+bx+c is (1, 7) and it passes through the point (3, 9). The function can be written as f(x) = a(x + 1)^2 + 7 => a(x^2 + 2x + 1) + 7 => ax^2 +...

Math
`3x^4 +2x^3 9x^2 12x 4 =` 0 We will factor `x^3` from the first 2 terms and 1 from the last three terms. `x^3(3x+2)  (9x^2 +12x +4) =` 0 Now we know that `(9x^2 + 12x +4)` is a complete...

Math
When 5000 is loaned at an interest rate of 10% per annum compounded annually, the principal increases every year by an amount equal to the interest earned on the principal in the last period. The...

Math
We have to determine 75/61 + 42/18 75/61 + 42/18 First equate the denominator of both the terms `(75*18)/(61*18) + (42*61)/(61*18)` => `1350/(61*18) + 2562/(61*18)` => `3912/(61*18)` =>...

Math
Find the slope of the line through the points (2,5) and (1,4). `m=(y_2y_1)/(x_2x_1)=(54)/(21)=1/3`

Math
I will assume that the base of the tissue is a square with the length of the side is 8.5 and the height is 7.7 First we will find the volume: ==> V = L X W X H = 8.5 X 8.3 X 7.7 = 556.325 cubic...

Math
Find the vertical asymptotes for `f(x)=(x+8)/(x^2100)` . Note that vertical asymptotes occur when a function with no common factors in the numerator and denominator has a factor that causes a...

Math
`6x^5 51x^3 27x` `` First we will factor 3x . `==gt 3x( 2x^4 17x^2 9)` `` Now we will factor the quadratic equation. `==gt 3x(2x^2 +1)(x^29) ` `==gt 3x(2x^2+1)(x3)(x+3)`

Math
Given the expression: 27x^3 + 216 We need to solve for x. ==> 27x^3 + 216 = 0 We know that: `(a^3 + b^3)= (a+b)(a^2 ab + b^2)` `(27x^3 + 216)= ((3x)^3 + 6^3) ` `==gt a = 3x ==gt b= 6 ` `==gt...

Math
`(y^364)/(3y^2 17y +20)` First we will factor the numerator and denominator. ==> We know that `(a^3b^3)= (ab)(a^2+ab+b^2) ` `==gt (y^364)= (y4)(y^2 +4y +16)` `` ==> Now we will factor...

Math
3a(x+5)= 114 given that a = 4 ==> 3*4 (x+5)= 114 ==> 12(x+5)= 114 Now we will divide by 12. ==> (x+5)= 114/12 = 19/2= 9.5 ==> x = 9.5 5= 4.5 ==> x = 4.5

Math
(3+i) + (73i)= We will expand brackets. ==> 3 i + 7  3i = Now we will combine like terms. ==> (3+7) + (i3i) = ==> 4 + (4i) Then, the final form of the complex number is as...

Math
If you have 11 counters you can only form 1 array: 1x11. With ten counters, you can form a 1x10 array, as well as a 2x5 array. Thus even though 11>10, you can form more arrays with ten counters...

Math
The figure shows that the height of the cylinder is equal to the diameter of the sphere. The volume of the cylinder is V = Area of the base*height The base of cylinder is a circle. The radius of...

Math
You need to find the coefficient n if f = (x+1)g. Replacing f and g by the corresponding equations yields: `x^3nx+1 = (x+1)(x^2x+1)` You need to open the brackets to the right side: `x^3 n x+1 =...

Math
A tribonacci sequence can be defined as `t_n=t_(n1)+t_(n2)+t_(n3)` for `n>4` or as a,b,c,a+b+c,a+2b+2c,2a+3b+4c,... Here we have 5,11,_,_,_,_,_,260,... Let the third element be x; then the...

Math
You need to solve the integral `int (xdx)/(x+1)` Adding and subtracting 1 to numerator yields: `int ((x + 1  1)dx)/(x+1) = int ((x+1)dx)/(x+1) int dx/(x+1)` `int (xdx)/(x+1) =int dx  ln x+1 +...

Math
You need to know that the rational number `pi` expresses the relation between the circumference and the diameter of a circle. I suggest you to try the following experiment. Calculate the...

Math
You should come up with the following substitution such that: `e^x = t =gt e^x dx = dt` . If you change the variable, you should calculate the limits of integration. `x = 0 =gt e^0 = 1 = t` `` `x =...

Math
The given equation is: Ax = CBy ==> By = Ax + C In order to rewrite in the slope form, we need to isolate y on one side and determine the coefficient of x which is the slope. ==> To isolate...

Math
`d/dx log x^2` Chain rule: d(inside)*d(outside with inside unchanged) `2x*1/(x^2ln10)` =`2/(xln10)`  Alternatively you could have used a log rule to pop the 2 out front in the...

Math
It is given that: 2x^3+3x^3y^2 = 4y^2+1. To determine `(dy)/(dx)` use implicit differentiation. (2x^3+3x^3y^2)' = (4y^2+1)' => `6x^2 + 3x^3*2y*(dy)/(dx) + 9x^2*y^2 = 8y*(dy)/(dx)` =>...

Math
`7x^3 + 2x^2 +28x + 8 = 0` `` First we will rearrange terms. `(7x^3 + 28x) + (2x^2 + 8) = 0` `` Now we will facotor each terms. Factor 7x from the first two terms and 2 from the last two terms....

Math
The solution of the given linear equations is: `x/14  1/2 = 1/7` => `x/14 = 1/7 + 1/2` => `x/14 = (72)/14` => `x = 5` `(4/5)x + 3 = 2x  3/5` => `2x  (4/5)x = 3 + 3/5` => `1.2x...