# Tutor profile: Carolyn J.

## Questions

### Subject: Pre-Calculus

Using the Laws of Exponents, simplify the following expression: (x^3)^6 + (y^7)(y^4)

There are two important exponent rules to know and apply here. They are: 1. ((x^a)^b) = x^ab 2. (x^a)(x^b) = x^(a+b) In the problem given, the first portion matches rule 1. When a variable with an exponent, is raised to another exponent, the exponents are simply multiplied together. Since 3x6 = 18, then the first variable is simply x^18. Next, the second portion matches rule 2. When a variable with an exponent is multiplied by the SAME variable, raised to another exponent, then the exponents can be added. Therefore, since 7+4 = 11, then the second variable is simply y^11. Therefore, the entire expressions simplifies to: x^18 + y^11

### Subject: Calculus

Using a derivative, find the point at which the slope of the graph of the following equation is zero: y(x) = x^2 + 5x + 2

It is known that the derivative of an equation represents the slope. Therefore, by taking the derivative of the equation, the slope of its graph at any point can be determined. It can be helpful to consider each variables derivative separately: i.e. d/dx(y(x)) = d/dx(x^2 + 5x + 2) = d/dx(x^2) + d/dx(5x) + d/dx(2) First, d/dx(x^2) is determined using the exponent rule. The exponent, 2, is dropped down and multiplied by the variable before the x, which is 1. Also, the exponent is reduced by one, therefore the 2 exponent becomes 1. So, the derivative is d/dx(x^2) = 2x. Next, d/dx(5x) is determined. Since the exponent of the x variables is 1, then 1 is multiplied by the value in front, 5, to give a value of 5. Then, the same as before, one is subtracted by the exponent. Then, x is to the zero power. Since anything to the zero power is simply one, this derivative reduces to 5. Finally, for d/dx(2), the derivate of a constant value is always zero. Then, the derivate of the whole equation becomes: 2x + 5 This derivate represents the slope of the given equation at various points. Since we want to know when the slope is zero, simply set this equation equal to zero: 2x + 5 = 0. The result is x = -2.5. At this point, the slope of the graph is zero. Visually, this is represented as the bottom of a parabola.

### Subject: Algebra

The following equation represents the profits of a lemonade stand, where "y" is the amount of $ made, and "x" is the amount of lemonades sold: y = 3x + 5 If the lemonade stand has a goal of earning $50 by the end of the day, how many lemonades must be sold?

Notice that "y" represents the amount of $ made. Since the goal is to make $50, this value can be used for "y". Then, the equation becomes: 50 = 3x + 5 Next, a value of 5 can be subtracted from both sides. This results in: 45 = 3x Finally, each side can be divided by 3. This results in: x = 15 Since "x" represents the amount sold, then 15 lemonades must be sold to reach the goal.