
Math
The equation 21 log(3) cube root of x + log(3)9x^2 log(3)9 = 2 has to be solved. 21 log(3) cube root of x + log(3)9x^2 log(3)9 = 2 can be written as 21 log(3) x^(1/3) + log(3) 9x^2  log(3) 9 =...

Math
Roots of a function f(x) are the values of x for which f(x) = 0. To find the roots graphically we plot the graph f(x) versus x and find the points where the graph intersects the xaxis. The value...

Math
Some of the properties of logs are: log(a*b) = log a + log b example: log 6 = log 3*2 = log 3 + log 2 log(a/b) = log a  log b example: log 5 = log(10/2) = log 10  log 2 log(a^b) = b*log a...

Math
The house decreases at the rate of 18% every year. If the initial value of the house was P and the rate of depreciation is r, the formula for the value after n years is P*(1  r)^n We need to find...

Math
The identity you had given to be proved was : (cos x + sin x)^2 + (cos x*sin x)^2 = 2 (cos x + sin x)^2 + (cos x*sin x)^2 = 2 opening the brackets gave => (cos x)^2 + (sin x)^2 + 2*sin x*cos x +...

Math
We have to prove that 1  (sin^6 x + cos^6 x) = 3(sin x)^2 (cos x)^2 Let's start with the left hand side. 1  [(sin x)^6 + (cos x)^6] => (sin x)^2 + (cos x)^2  (sin x)^6  (cos x)^6 => (sin...

Math
The identity csc x  cot x = sin x/(1+cos x) has to be proved. Start from the left hand side csc x  cot x csc x = 1/ sin x and cot x = cos x/ sin x => 1/sin x  cos x/sin x => (1  cos...

Math
The value of lim x>3 [sqrt ( x^2  9)/(x  3)] has to be found. lim x>3 [sqrt ( x^2  9)/(x  3)] substituting x = 3, gives the indeterminate form 0/0. => lim x>3 [sqrt (x  3)*...

Math
We need to find lim x > 2[ (x  2)/4  sqrt (4x  x^2)] substituting x = 2 , gives (0/4)  sqrt (8  4) => 0  sqrt 4 => 2 The required limit is 2.

Math
The value of lim x> 1[ (x  1)/sqrt(2x  x^2  1)] has to be found. It can be seen that a direct substitution of x = 1, yields 0/0 which cannot be determined. lim x>1[ (x  1)/sqrt(2x ...

Math
To simplify (8x^2  242) / (2x^2  25x+77) we have to factorize the numerator and the denominator and cancel any common factors. (8x^2  242) / (2x^2  25x + 77) => 2(4x^2  121) / (2x^2  14x ...

Math
To simplify (x^213x+42) / (14x^2+71x33) use the following steps: (x^213x+42) / (14x^2+71x33) factorize the denominator and the numerator => (x^2  7x  6x + 42) / (14x^2 + 77x  6x  33)...

Math
For a given data set, the range is the difference between the largest value and the smallest value. The mean is the sum of all the values divided by the number of values. The mode is the value that...

Math
The range for a data set is the difference between the largest value and the smallest value. In the case of the data set set you have provided: 94, 90, 88, 66, 94, 81, 102, 108, 88, the largest...

Math
Complex numbers are an extension to real numbers and extensively used in many aspects of mathematics relating to fields like engineering, electromagnetism, quantum physics, applied mathematics, and...

Math
log3 ( x+ 3) + log3 ( x5) = 2 First we will determine the domain, ==> x+ 3 > 0 and x5 > 0 ==> x > 3 and x > 5 ==> x > 5 is the domain.......(1) We will use...

Math
An exponential equation is one of the form y = a*b^x, where a is the initial value of y when x = 0 and it increases at a nonlinear rate. The value of y increases by the factor b, also called the...

Math
To determine the extremes of the function, we'll have to calculate the critical values of the function, that are the roots of the first derivative of f(x). We'll differentiate the function with...

Math
Sorry, the derivative is dy/dx=1/(cos2x+sin^2x)!

Math
We have to find the derivative of f(x) = (2x+1)^2+(2x2)^2 + 2(4x^22x2) f'(x) = [(2x+1)^2+(2x2)^2 + 2(4x^22x2)]' f'(x) = [(2x+1)^2]'+[(2x2)^2]' + [2(4x^22x2)]' Use the chain rule. f'(x) =...

Math
The vectors AB and CD are perpendicular to each other. In this case the dot product is equal to 0, as cos 90 = 0. The dot product of the vectors AB = (32a)*i+(a+1)*j and CD=(2a+1)*i+2*j is also...

Math
We have to prove : (1/(sin x^2x)+(1/cos^2x)=(tan x+(1/tan x))^2 Take the left hand side 1/(sin x)^2+ 1/(cos x)^2 => ((cos x)^2 + (sin x)^2)/(sin x)^2*(cos x)^2x => 1/(sin x)^2*(cos x)^2x...

Math
As it is tough to type theta, I will use "y" instead. It is given that :sin(2*pi  y)  sin(pi  y)  cos(3/2pi + y) = 3/2 sin (2*pi  y) = sin y sin (pi  y) = sin y cos (3*pi/2 + y) = sin y...

Math
The vectors u = m*i + (m+1)*j and v = 3*i + 5*j are collinear. This is possible if m / (m+1) = 3/5 m / (m+1) = 3/5 => 5m = 3(m + 1) => 5m = 3m + 3 => 2m = 3 => m = 3/2 The value of m = 3/2

Math
To simplify 8i/(2  2i) we have to convert the denominator to a real number. This can be done by multiplying it and the numerator by the complex conjugate or 2 + 2i 8i/(2  2i) => 8i(2 + 2i)/(2...

Math
The derivative of y is given as dy/dx = sqrt(4x^2+8x+12) y' = sqrt [ 4x^2+8x+12] => 2*sqrt [x^2 + 2x + 3] => 2*sqrt [ (x^2  2x + 1) + 4] => 2*sqrt [ (x  1)^2 + 4] y = Int[ y' dx]...

Math
We have to find the anti derivative of f(x)*cos x , where f(x) = ln(1 + (sin x)^2) Int [ f(x)*cos x dx] => Int [ ln(1 + (sin x)^2) * cos x dx] let sin x = y dy = cos x dx => Int [ ln(1 + y^2)...

Math
We need to find the indefinite integral of f(x)=3x^2+12x+18 Int [ f(x) dx ] => Int [ 3x^2+12x+18 dx ] => Int [ 3x^2 dx ] + Int[12x dx] + Int[ 18 dx ] => 3*x^3 / 3 + 12x^2 / 2 + 18x + C...

Math
The function f(x)=(x^2+2)^4 can be differentiated using the chain rule. According to the chain rule for f(x) = h(g(x)) f'(x) = h'(g(x))*g'(x) f(x) = (x^2+2)^4 take g(x) = x^2 + 2, h(x) = x^4 f'(x)...

Math
The equation that we have to find the solutions of is 4*2^(x^2)/2^(3x)=64 4*2^(x^2)/2^(3x)=64 => 2^(x^2)/2^(3x)=64/4 => 2^(x^2)/2^(3x)=16 => 2^(x^2) = 16*2^(3x) => 2^(x^2) = 2^4*2^(3x)...

Math
Let us find the solution of the equation 3^(2x6) = 81 3^(2x6) = 81 => 3^(2x  6) = 3^4 we can equate the exponent 2x  6 = 4 => x = 10/2 = 5 The number of elements in the set...

Math
We have to find the partial fractions of 2x/(x^29) 2x/(x^29) => 2x/(x+3)(x  3) => A/(x + 3) + B/(x  3) [A(x  3) + B(x + 3)]/(x + 3)(x  3) = 2x/(x+3)(x  3) => Ax  3A + Bx + 3B = 2x...

Math
Let f(x) = y y = sqrt(x  1) We'll interchange x by y: x = sqrt(y  1) We'll raise to square both sides, to eliminate the square root: x^2 = y  1 We'll add 1 both sides, to isolate y: x^2 + 1 =...

Math
First, we'll rewrite the expression of dy/dx, using the negative power rule: (2+cosx)^1 = 1/(2+cosx) dy/dx = 1/(2+cosx) => dy = dx/(2+cosx) To determine the primitive of the given function dy,...

Math
First, we'll create the function f(x) = sin x  x*cos x and we'll have to prove that f(x)>0. To study the behavior of the function, namely if it is an increasing or a decreasing function, we'll...

Math
First, we'll make the substitution sin x=t (t)^2  8*t + 12 = 0 We'll apply quadratic formula t1= [(8)+sqrt(6448)]/2 t1=(8+4)/2 t1=6 sin x=t1 <=> sin x=6 (impossible) Since the range of...

Math
We'll replace the function cot 45 by it's value 1. To simplify the difference, we'll transform it into a product. We'll have to express the value 1 as being the function cosine of an angle, so...

Math
To solve the equation dy/dx = 0, we'll have to differentiate y with respect to x. dy/dx = (11x^422x^2+33)' dy/dx = 11*4*x^3  22*2*x + 0 dy/dx = 44x^3  44x We'll cancel dy/dx: dy/dx = 0 <=>...

Math
First, we'll apply the rule of negative power for the term x^2: x^2 = 1/x^2 To get the squares x^2 and (1/x^2), we'll raise to square the expression x (1/x). If we want to raise to square x...

Math
We'll determine the possible values of x, for the square root to exist. For this reason, we'll impose the following constraint: the radicand has to be positive or, at least, zero. 2x + 1 >= 0...

Math
To evaluate the indefinite integral of f(x)=sin5x*cos3x, we'll apply the formula to transform the product of trigonometric functions into a sum. We'll use the formula: sin a * cos b =...

Math
We'll use the binomial raised to cube identity: (a+b)^3 = a^3 + b^3 + 3a*b*(a+b) In this case, a = x1 and b = x2 We'll replace a and b by x1 and x2 and we'll have: (x1+x2)^3 = x1^3 + x2^3 +...

Math
We have to solve: (cos x)^2  (sin x)^2 + sin x = 0 (cos x)^2  (sin x)^2 + sin x = 0 substitute (cos x)^2 = 1  (sin x)^2 => 1  (sin x)^2  (sin x)^2 + sin x = 0 => 1  2*(sin x)^2 + sin x...

Math
Let the terms of a geometric progression be : a1, a2, a3, a4 Given that a3= 1 and the common difference is 1/2. ==> Then we know that: a3 = a1*r^2 1 = a1* (1/2)^2 1= a1/ 4 ==> a1= 4 ==>...

Math
Let us assume that the number is x. Then 3 times square x is 3x^2. And we will add 2 times fifth x ==> 3x^2 + 2x/5 = 3x^2 +(2/5)x ==> All divide by (x+7) ==> [(3x^2 + (2/5)x] / (x+7)...

Math
sqrt(x^2 + 3x 5) = x+ 2 First we will square both sides. ==> x^2 + 3x 5 = (x+2)^2 ==> x^2 + 3x 5 = x^2 + 4x +4 ==> 3x 5 = 4x+4 ==>. x = 9 Let us check. ==> sqrt( 81275) =...

Math
Let the first speed be S1 = 55 mph and the time is T1 = 24 min. The second speed is S2= 135 and the time is T2 = 135 min. We need to find the total distance . Let the total diostance be D = D1 +...

Math
Given that : log a = 12 log b= 3 We need to find the value of the expression : log (a^2*b^3)^6 First we know that log a^b= b*log a. ==> log (a^2*b^3)^6 = 6*log (a^2*b^3) Now we know that log ab...

Math
log3 (12n1)  log3 (2x+7) = 2 First we will find the domain. ==> 12n1 > 0 ==> n> 1/12 ==> 2n+7 > 0 ==> n > 7/2 Then The domain is n > 1/12 .............(1) Now we will...

Math
81^2x = 27^(4x1) First we will simplify by factoring the bases. ==> 81 = 3*3*3*3= 3^4 ==> 27 = 3*3*3 = 3^3 Now we will substitute. ==> 3^4^2x = 3^3^(4x1) But we know that x^a^b= x^ab....