Exponents are numerals that indicate an operation on a number or variable. The interpretation of this operation is based upon exponents that symbolize natural numbers (also known as positive integers). Natural-number exponents are used to indicate that multiplication of a number or variable is to be repeated. For instance, 5 × 5 × 5 is written in exponential notation as 53 (read as any of "5 cubed," "5 raised to the exponent 3," or "5 raised to the power 3," or just "5 to the third power"), and x × x × x × x is written x4. The number that is to be multiplied repeatedly is called the base. The number of times that the base appears in the product is the number represented by the exponent. In the previous examples, 5 and x are the bases, and 3 and 4 are the exponents. The process of repeated multiplication is often referred to as raising a number to a power. Thus the entire expression 53 is the power.
Exponents have a number of useful properties:
Any of the properties of exponents are easily verified for natural-number exponents by expanding the exponential notation in the form of a product. For example, property number (1) is easily verified for the example x3x2 as follows:
Property (5) is verified for the specific case x2y2 in the same fashion:
Exponents are not limited to the natural numbers. For example, property (3) shows that a base raised to a negative exponent is the same as the multiplicative inverse of (1 over) the base raised to the positive value of the same exponent. Thus 2-2 = 1/22 = 1/4.
Property (6) shows how the operation of exponentiation is extended to the rational numbers. Note that unit-fraction exponents, such as 1/3 or 1/2, are simply roots; that is, 125 to the 1/3 power is the same as the cube root of 125, while 49 to 1/2 power is the same as the square root of 49.
By keeping properties (1) through (6) as central, the operation is extended to all real-number exponents and even to complex-number exponents. For a given base, the real-number exponents map into a continuous curve.
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