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First take natural logs of both sides, giving us: ln(3^x * e^4x) = ln(e⁷).  The RHS can be simplified using the definition of natural logs, so ln(e⁷) = 7.

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For the first part, use the 'complete the square method', which gives (x-7/2)² -49/4+2, simplified to (x-7/2)²-41/4. To find the answer for the second part, set x=7/2 so that the bra...

We know thst the binomial expansion formula is (1+x)ⁿ = 1+nx +(n)(n-1)(x²)/2! ...etc

Thus we want to rearrange the expression we are given into the form (1+x)ⁿ so t...

Firstly, the (2x(^3) - 7)/(3(sqrt(x))) can be split into (2x(^3))/(3(sqrt(x)) and -7/(3(sqrt(x)). These can then be simplified to (2/3)x(^5/2) and -(7/3)x(^-1/2) respectively. This then gives the equation...

First, we need to separate the RHS as components of U and V. Using the LATE (logarithms, algebra, trigonometry, exponentials) technique, we see that logarithms have priority to algebra hence U = lnx and d...