Simplify: m9 128a5 (c) V-24x5 (d) V65. (a) (b) All variables represent positive real numbers. We will factor each radicand into two factors, one of which is a perfect nth power. We can then apply the rule the nth root of a product is the product of the nth roots to simplify the radical expression. (a) The greatest perfect-square factor of m° is m8. m9 = m8. m Write m° in factored form as m8 - m. = V m8 Vm Use the product rule for radicals. 8 Vm Simplify V m8. (b) Since the greatest perfect-square factor of 128 is 64 and the greatest perfect-square factor of a5 is a“, the largest perfect-square factor of 128a5 is 64 We write 128a as 64a* - 2a and proceed as follows: 128a5 = V 64a · 2a Write 128a5 in factored form as 64a - 2a. = V 64a4 . V 2a Use the product rule for radicals. = 8a? /2a Simplify V64a4. (c) We write -24x as -8x - 3x² and proceed as follows: V-24x5 = V-8x³ - 3x² 8x is the largest perfect-cube factor of 24x. Since the radicand is negative, we factor it using -8x. = V-8x3 · V3x² Use the product rule for radicals. 8 V3x Simplify V-8x³.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
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