Results 1 to 2 of 2

Math Help - Arithmetic

  1. #1
    Super Member
    Joined
    Jun 2008
    Posts
    829

    Arithmetic

    If 2^{2a}-2^{2b}=55..
    What is the value of a and b?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,735
    Thanks
    642
    Hello, Apprentice123!

    If 2^{2a}-2^{2b}=55, what is the value of a and b?
    With one equation in two unknowns, there are infinite solutions.
    . . I'll show you two of them . . .


    We have: . (2^a-2^b)(2^a + 2^b) \:=\:55



    Let: . \begin{array}{cccc}2^a-2^b &=& 5 & [1] \\ 2^a + 2^b &=& 11 & [2] \end{array}

    Add [1] and [2]: . 2\!\cdot\!2^a \:=\:16 \quad\Rightarrow\quad 2^a \:=\:8 \quad\Rightarrow\quad a \:=\:3

    Subtract [2] - [1]: . 2\!\cdot\!2^b \:=\:6 \quad\Rightarrow\quad 2^b \:=\:3 \quad\Rightarrow\quad b \:=\:\log_2\!3

    . . Therefore: . (a,b) \;=\;(3,\:\log_2\!3)



    Let: . \begin{array}{cccc}2^a - 2^b &=& 1 & [3] \\ 2^a + 2^b &=& 55 & [4]\end{array}

    Add [1] and [2]: . 2\!\cdot\!2^a \:=\:56\quad\Rightarrow\quad 2^a \:=\:28 \quad\Rightarrow\quad a \:=\:\log_2\!28

    Subtract [2] - [1]: . 2\!\cdot\!2^b \:=\:54 \quad\Rightarrow\quad 2^b \:=\:27 \quad\Rightarrow\quad b \:=\:\log_2\!27

    . . Therefore: . (a,b) \;=\;(\log_2\!28,\:\log_2\!27)

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. A sum on arithmetic mean...
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 30th 2010, 02:47 AM
  2. Arithmetic Progression or Arithmetic Series Problem
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: October 8th 2009, 12:36 AM
  3. arithmetic
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 26th 2009, 10:24 AM
  4. Arithmetic Mean
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 10th 2007, 03:50 PM
  5. Arithmetic Mean
    Posted in the Algebra Forum
    Replies: 1
    Last Post: November 8th 2006, 07:01 AM

Search Tags


/mathhelpforum @mathhelpforum