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Ahmed E.

Maths & Science Tutor(5000+ hours/tutoring) (WebAssign expert)

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Basic Math

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Question:

Solve for x x+6=8

Ahmed E.

Answer:

To solve the linear equation you need to keep the variable in a side and all the other constants/numbers in the other side and to do that in this problem so we need to cancel the 6 by adding -6 to both sides it becomes x+6-6=8-6 >>>> x=2

Calculus

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Question:

Derive the equation y=5x(x^2+2)^3

Ahmed E.

Answer:

It is very clear that those are 2 equations multiplied together f=5x g=(x^2+2)^3 so we will use the product rule to get the derivative of this funciton ( f ⋅ g ) ′ = f ′ ⋅ g + f ⋅ g ′ g function is a polynomial function so we wil use the power function rule which is g'(x) =n*g(x)^n−1 * dx g=(x^2+2)^3 >>> d(x^2+2)=2x >> g'= 3(x^2+2)^2 *2x= 6x(x^2+2)^2 f=5x >>>> f'=5 so (f.g)'= 5*(x^2+2)^3 +5x*6x(x^2+2)^2 =5(x^2+2)^3 +30x^2(x^2+2)^2 it could be simplified by taking 5(x^2+2)^2 as a common factor >>>>> 5(x^2+2)^2 (x^2+2+6x) it might be expanded by expanding the quadratic equation and then multiplying it bt (x^2+6x+2)

Algebra

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Question:

Solve for x: 3^x= 27

Ahmed E.

Answer:

This is an exponential equation : you may go through 2 tracks to solve such a problem. 1st track: You may try to change the right hand side part to be something in the form of 3^number :the valie of this number will be equal the value of x , for this problem 27 will be 3^3 so 3^x=3^3 so if we campared both sides it will be very obvious that x=3 2nd track: This track is more general and easier, as this is an exponential equation and we need to know the valie of the exponent so we may take the natural logarithm (ln) for both side to change the poition of x from the exponent to be infront of the number 3^x=27 so after taking ln for both side, ln3^x=ln27 using the properties of natural logarithsm >>> x*ln3=ln27 so now we need to divide both sides by the coefficient of x which is ln3 so x*ln3/ln3=ln27/ln3 >>>> x=ln27/ln3 >>>>> x=3

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