it is shown that the number of multiplication needed to compute x100 is 8. Using the same method, how many multiplications are needed to compute x111? Example 1.1.1: Exponentiation// Let us remind you that if n is an integer greater than one then x" denotes the// product of n x's; x' is x itself; x° equals 1; and, if x + 0 then x// Then (x") x (x") = x" + " and (x")" = x"n for all integers m and n.denotes 1/x".Evaluating x'100would seem to require 99 multiplications, but it can be done withfar fewer as follows:multiply xmultiply x? by x²multiply xmultiply x by x3multiply x16 by x16 to obtain x32multiply x2 by x2 to obtain x64.to obtain x?to obtain x4to obtain xto obtain x16by xby x4Now multiply x4 by x" to obtain x6 and multiply x by x* to obtain x100.The total number of multiplications was only 8.// How many multiplications would be required to evaluate x23, x204, or x2005?The first stage of the algorithm used 6 squaring operations to obtain x?for j = 1, 2, ..6, and the second stage used the fact that100 = 64+ 32 +4 = 26 +25 + 22.Do you think that every positive integer can be expressed as a sum of certain,distinct powers of 2? If we were to use RPM to multiply 1 times 1,349, we would getAВ113492674433716816843242642112810256551221024113492°, then the rest of the A-values areWhen A is given the initial value 1consecutive powers of 2, and the product returned by RPM is the initial value of Bexpressed as a sum of distinct powers of 2.%3D

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it is shown that the number of multiplication needed to compute x100 is 8. Using the same method, how many multiplications are needed to compute x111?

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

it is shown that the number of multiplication needed to compute x¹⁰⁰is 8. Using the same method, how many multiplications are needed to compute x¹¹¹?

Example 1.1.1: Exponentiation
// Let us remind you that if n is an integer greater than one then x" denotes the
// product of n x's; x' is x itself; x° equals 1; and, if x + 0 then x
// Then (x") x (x") = x" + " and (x")" = x"n for all integers m and n.
denotes 1/x".
Evaluating x'
100
would seem to require 99 multiplications, but it can be done with
far fewer as follows:
multiply x
multiply x? by x²
multiply x
multiply x by x3
multiply x16 by x16 to obtain x32
multiply x2 by x2 to obtain x64.
to obtain x?
to obtain x4
to obtain x
to obtain x16
by x
by x4
Now multiply x4 by x" to obtain x6 and multiply x by x* to obtain x100.
The total number of multiplications was only 8.
// How many multiplications would be required to evaluate x23, x204, or x2005?
The first stage of the algorithm used 6 squaring operations to obtain x?
for j = 1, 2, ..
6, and the second stage used the fact that
100 = 64+ 32 +4 = 26 +25 + 22.
Do you think that every positive integer can be expressed as a sum of certain,
distinct powers of 2? If we were to use RPM to multiply 1 times 1,349, we would get
A
В
1
1349
2
674
4
337
168
16
84
32
42
64
21
128
10
256
5
512
2
1024
1
1349
2°, then the rest of the A-values are
When A is given the initial value 1
consecutive powers of 2, and the product returned by RPM is the initial value of B
expressed as a sum of distinct powers of 2.
%3D

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