Square and Cube Root

There may be more than two identical factors of a number. You know that 2X4 = 8. Are there any other factors of the number 8 It is the product of 2 X 2 X 2. A cube root is one of three identical factors of a number. Thus, what is the cube root of 8 It is 2. A cube root also is indicated by a radical. 

Check your calculator handbook for more information on finding roots. 

The estimated digit must be read between the millimeter markings on the top ruler. Why is the bottom ruler less precise  

Measuring tools are never perfect, nor are the people doing the measuring. Therefore, whenever you measure a physical quantity, there will always be uncertainty in the measurement. The number of significant figures in the measurement indicates the uncertainty of the measuring tool. 

When a measurement is provided, the following series of rules will help you to determine how many significant figures there are in that measurement. 

Determine the cube root of 16 using your calculator. 

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Except in simple cases (square and cube roots) radical signs are replaced by fractional exponents. If n is odd,... 

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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. 

Earlier, perfect squares and cubes were described. For example, 64 is a perfect square because 8 x 8, or 82 equals 64. This factor, 8, is called the square root of 64. Likewise, since 3 x 3 x 3, or 33 equals 27, the cube root of 27 is 3. 

Square root molecular wmg t (MW2), defined as MW2 = MW, and cube root molecular weight (MW3), defined as MW3 = MW and corresponding to a linear dimension of size, are also used as molecular size descriptors. 

Higher roots are defined in the same way as square roots and cube roots. 

Two other historical asides about this result are interesting. First, the dimensionless quantities b and b suggested by Reynolds were renamed y-factors by Chilton and Colburn. These factors are common in the older literature, especially as Jd and Jh. Second, the exponent of on the Schmidt and Prandtl number is frequently subjected to theoretical rationalization, especially using boundary-layer theory. Chilton is said to have cheerfully conceded that the value of was not even equal to the best fit of the data, but was chosen because the slide rules in those days had square-root and cube-root scales, but no other easy way to take exponents. 

Figure 1. Graph of the Reciprocal of the Diffusivity against the Product of the Cube Root of the Molar volume and the Square Root of the Molecular Weight...

It is seen that if the diffusivity is to be correlated with the molecular weight, then a knowledge of the density of the solute is also necessary. The result of the correlation of the reciprocal of the diffusivity of the 69 different compounds to the product of the cube root of the molecular volume and the square root of the molecular weight is shown in Figure 1. A summary of the errors involved is shown in Figures 2 and 3... 

Roots of exponentials are obtained in a similar way. To extract a square root, use the Vx key. For other roots, use they key. To obtain a cube root, enter the number, press they key, enter 1/3 (0.333333333) and read the answer ... 

In both equations, k and k are proportionality constants and 0 is a constant known as the critical exponent. Experimental measurements have shown that 0 has the same value for both equations and for all gases. Analytic8 equations of state, such as the Van der Waals equation, predict that 0 should have a value of i. Careful experimental measurement, however, gives a value of 0 = 0.32 0.01.h Thus, near the critical point, p or Vm varies more nearly as the cube root of temperature than as the square root predicted from classical equations of state. 

In the early days of water radiolysis, it was empirically established in several instances that the reduction of molecular yield by a scavenger was proportional to the cube root of its concentration (Mahlman and Sworski, 1967). Despite attempts by the Russian school to derive the so-called cube root law from the diffusion model (Byakov, 1963 Nichiporov and Byakov, 1975), more rigorous treatments failed to obtain that (Kuppermann, 1961 Mozumder, 1977). In fact, it has been shown that in the limit of small concentration, the reduction of molecular yield by a scavenger should be given by a square root law in the orthodox... 

The number 64 has another square root, namely -8, since -8 x -8 = (-8)2 = 64. If we want to indicate the positive square root, use the fn radical symbol to denote square root. In an expression such as V25, 25 is called the radicand, and the expression is the radical. So V64 = 8, and -V64 = -8. The notation for cube root is Vm. 

It is interesting that iL is independent of the viscosity of the solution note, however, that this is only true for high Schmidt numbers (thin diffusion layer). Also, iL is dependent only on the cube root of the fluid flow (cf. square root at the RDE). This lower sensitivity can result in larger errors in the determination of mass transfer and kinetic parameters. 

Another way of expressing QD values is to state them as the cube root of the expl wt because certain detonation phenomena scale according to a cube root law. One of these is the instantaneous peak overpressure with distance (Ref 11). Damage can be related to overpressure by the cube root law except with respect to damage within inhabited structures and with respect to flying debris, for both of which a square root law is more nearly correct. 

Figure 2.22 Different fits for the relationship between reflectance and lightness. Note that the axes have been switched, compared with Figure 2.21. Logarithmic (excluding zero), cube root and square root fits are shown for Land s experimental data. The data is best approximated by the function Y = 9.033/J0 4296 — 0.0151, which is also shown.

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. 

From Eq. (3.1) we see that the Debye frequency iw varies proportionally to the velocity of elastic waves, divided by the cube root of the volume, and from Eq. (3.9) wc see that the velocity of either longitudinal or transverse waves varies inversely as the square root of the compressibility times the density, if we assume that Poisson s ratio is independent of the volume. As we shall see later, this assumption can hardly be... 

Fig. 9a-d. Correlations of volume (S) of sorbed benzene per gram of (Sty)[ x (DVB), particles (80% by weight) enmeshed in PTFE microfibers with (a) the cross-link density, 1 /X (where X, equal to 1/x when x is <0.1, is the number of backbone carbon atoms between crosslink junctions), (b) the reciprocal of crosslink density (X), (c) the square root of X, and (d) the cube root of X. 

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