Powers, Exponents, and Roots

How did you do on the powers, exponents, and roots Benchmark Quiz Check your answers here, and then analyze your results to figure out your plan of attack to master these topics. 

Exponents are the method for raising numbers to different powers. Exponents and roots are less likely to be encountered in everyday life, but they are very important in fields such as science, computer programming, and applied mathematics. 

Integral Exponents (Powers and Roots) If m and n are positive integers and a, b are numbers or functions, then the following properties hold ... 

Any base number raised to the second power is called the square of the base. So 42 is said to be four squared. Since 42 = 4 x 4, which is 16, 16 is called a perfect square. Any base number raised to the third power is called the cube of the base. So 43 is said to be four cubed. Since 43 = 4 x 4 x 4, which is 64, 64 is called a perfect cube. It is helpful to know some of the perfect squares and cubes, both for raising to an exponent, and taking roots, discussed in a later section. 

Powers and Roots When numbers expressed in exponential notation are raised to a power, the exponents are multiplied by the power. When the roots of numbers expressed in exponential notation are taken, the exponents are divided by the root. 

In summary, the sensitivity and resolving power of flow-driven planar FAIMS at moderate ion current scale as the inverse exponent and square root of separation time ties, respectively. These laws allow predicting many aspects of analyses without simulations and permit their validation when simulations are needed. The variation of ties via adjustment of flow rate is an effective practical approach to control of planar FAIMS resolution. In strongly curved gaps, ties is less important and other instmmental parameters make a greater difference (4.3). 

The exponents a and p are called critical exponents. We discuss power laws and critical exponents on polymer configurations in the next chapter. Here we point out that all these laws have their roots in the concept of thermodynamic equilibrium. Consider the equation... 

Sedimentation experiments also provide evidence of non-asymptotic behavior for semi-dilute solutions in a good solvent. Power law fits to the data provide exponents varying from 0.59 to 0.82, while the theoretical scaling exponent is 0.54. These exponents are consistent with those obtained for Dcoop- detailed review on the cooperative diffusion and sedimentation results has been recently published by Nystrom and Roots. 

When an exponent is a fraction, the denominator of this fractional exponent means the root of the base number, and he numerator means a raise of the base to that power ... 

When the exponent refers to a root rather than a power i.e. y — xl v, we use the fact that the derivative is defined in terms of a ratio and consider the root as the inverse function of the power yq. Let us illustrate this for the square root function y — y/x, ori = y2. We calculate first ... 

For a number with a fractional exponent, the numerator of the exponent tells you the power to raise the number to, and the denominator of the exponent tells you the root you take. 

I. To extract the square root of a power of 10, divide the exponent by 2. If the exponent is an odd number it should be increased or decreased by 1, and the coefficient adjusted accordingly. To extract the cube root of a power of 10, adjust so that the exponent is divisible by 3 then divide the exponent by 3. The coefficients are treated independently. 

Modulo a prime power, pL where p 2, a number y is a quadratic residue if and only if it is one modulo p. Furthermore, each quadratic residue has two square roots again. This can be seen by considering the isomorphism with the additive group modulo 0 = (p - l)p If g is a generator, exactly the elements with an even exponent e are the quadratic residues, and g and are the roots. 

Simulation methods construct the wavefunction (or at positive temperature the /V-body density matrix) by sampling it and therefore do not need its value everywhere. The complexity then usually has a power-law dependence on the number of particles, T< N, where the exponent typically ranges from 1 s 5 4, depending on the algorithm and the property. The price to be paid is that there is a statistical error which decays only as the square root of the computer time, so that T e. ... 

This is applicable to various carrier symmetries such as planar, cylindrical and spherical. Here Q is the amount of molecules released per unit exposed area of the carrier, t denotes time and a, b and k are constants. This power-law function is related to the Weibull function that has been suggested as a universal tool for describing release from both Euclidian and fractal systems, and may be considered as a short-time approximation of the latter (Kosmidis et al. 2003). The constant a takes initial delay and burst effects into account, and is a kinetic constant (Jamzad et al. 2005). The power law exponent, k, also called the transport coefficient, characterises the diffusion process and equals 0.5 for ordinary case I (or carrier conttoUed) diffusion in systems for which no swelling of the carrier material occurs, which can be expected for mesoporous material (Ritger and Peppas 1987). Diffusion-controlled release from a planar system, in which the carrier structure is inert, may be described by the Higuchi square-root-of-time law ... 

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