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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?

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 & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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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?

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
Transcribed Image Text: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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