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which is de ned to mean that B C2 Likewise, B1=3 is the cube root of B, which can be written in the form p B1=3 3 B C meaning that and so on. Thus 16 1=2 8 23 . All of the foregoing operations with exponents can be summarized in the following laws of exponents, which are valid for all positive and negative integral and fractional values of the exponents. 1. 2. 3. 4. B a B b B a b (exponents add in multiplication) B a =B b B a b (numerator exponent minus denominator exponent in division) B a b B ab (B to power a, raised to power b) B=C a B a =C a (fraction B/C raised to power a) B C3 p p 16 4, because 4 2 16, and 8 1=3 3 8 2, since
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In the above, note that (1) and (2) apply for like base numbers only. To close our discussion of exponents, suppose a 0 in law (1) above; for this case, using law (1), we have that B 0 Bb B 0 b Bb , which can be true only if B 0 1. Thus it is de ned that B to the zero power is one, that is, B 0 1, where B has any nite value except zero (because no value is assigned to the expression 00 ). The right triangle is important in algebraic applications. Let us denote the sides of a right triangle by a, b, and c, where side c, opposite the 908 angle, is the hypotenuse, as shown below. In any right triangle it is true that c2 a2 b2 the Pythagorean theorem thus, c p a2 b2
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The degree of an algebraic equation is equal to the highest power of the unknown in the equation. Thus, if a and b are known constant values, and x denotes the value of an unknown quantity, the basic FIRST DEGREE or linear equation is of the form ax b 0 the solution of which is x b=a Or, if a, b, and c are known constant values, with x denoting the value of an unknown quantity, the basic SECOND DEGREE or quadratic equation is of the general form ax2 bx c 0 the solutions of which are x p 1 b b2 4ac 2a
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In closing this item, let us be reminded that DIVISION BY ZERO IS NOT PERMITTED in algebra. The reason for this restriction can be shown as follows. Let A, B, and C represent three numbers, thus A C; meaning that A BC B If, however, B 0, then A 0 C, which means that A 0 regardless of the value of 0 C; thus, setting B 0 leads to the expression C, which is meaningless, since C has no 0 de nite value.
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Note 2.
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Fundamental Units
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The four fundamental quantities in the mks system are LENGTH; l; measured in meters m MASS; m; measured in kilograms kg TIME; t; measured in seconds s ELECTRIC CHARGE; q; measured in coulombs C 1 meter 39:37 inches 3:281 feet 1 kilometer 1000 meters 0:6214 miles 1 kilogram 1000 grams 2:205 pounds In the mks system force is measured in NEWTONS, where 1 newton 0:2248 pounds of force 1 pound of force 4:448 newtons In physics, ENERGY is measured in terms of the ability to do work; in the mks system the basic unit of energy is the JOULE, where 1 joule 0:7376 foot-pounds of work 1 foot-pound of work 1:356 joules We note that heat is a form of energy; thus, mechanical and electrical energy can be transformed into heat, and heat can be transformed into mechanical and electrical energy. Heat is measured in calories , where 1 calorie 4:186 joules of work 3:0876 foot-pounds of work To conclude, POWER is the time rate of doing work; the UNIT OF POWER in the mks system is the watt, where
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