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When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base. [2] Thus 3 + 5 2 = 28 and 3 × 5 2 = 75. These conventions exist to avoid notational ambiguity while allowing notation to remain brief. [4]
When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, 3 5 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.
For n greater than about 4 this is computationally more efficient than naively multiplying the base with itself repeatedly. Each squaring results in approximately double the number of digits of the previous, and so, if multiplication of two d -digit numbers is implemented in O( d k ) operations for some fixed k , then the complexity of ...
Each of the operations above are defined by iterating the previous one; [1] however, unlike the operations before it, tetration is not an elementary function. The parameter is referred to as the base, while the parameter may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for ...
When multiplication is repeated, the resulting operation is known as exponentiation. For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 2 3, a two with a superscript three. In this example, the number two is the base, and three is the exponent. [26]
"In essence, this money has been stolen from all of us for all these years," said an 84-year-old woman whose late husband's Social Security benefits were slashed. "It's not fair."
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