math

How (1+4x^-6) equal x^-6 +4?

  1. 👍 0
  2. 👎 0
  3. 👁 150
  1. 1+4x^-6 = x^-6+4
    3x^-6 = 3
    x^-6 = 1
    x^6 = 1.
    Take the 6th root of both sides:
    X = 1.

    1. 👍 0
    2. 👎 0
  2. x = ±1 since all our powers are even.

  3. 1 + 4 x⁻⁶ = x⁻⁶ + 4

    1 + 4 / x⁶ = 1 / x⁶ + 4

    Subtract​ 1 from both sides:

    4 / x⁶ = 1 / x⁶ + 3

    Multiply both sides by x⁶:

    4 = 1 + 3 x⁶

    Subtract​ 1 from both sides:

    3 = 3 x⁶

    Divide both sides by 3:

    1 = x⁶

    Subtract​ 1 from both sides:

    0 = x⁶ - 1

    x⁶ - 1 = 0

    Substitution:

    x² = u

    x⁶ = u³

    x⁶ - 1 = 0

    u³ - 1 = 0

    u³ - 1³ = 0

    Difference of cubes:

    a³ - b³ = ( a - b ) ( a² + a ∙ b + b² )

    In this case:

    u³ - 1³ = ( u - 1 ) ( u² + u ∙ 1 + 1² )

    u³ - 1³ = ( u - 1 ) ( u² + u + 1 )

    This expression will be equal to zero when both terms are equal to zero.

    ( u - 1 ) = 0 and u² + u + 1 = 0

    u - 1 = 0

    Add 1 to both sides:

    u = 1

    x² = 1

    x = ± √1

    x = ±1

    The solutions are:

    x = - 1 and x = 1

    u² + u + 1 = 0

    Try to solve that.

    The solutions are:

    u = - 1 / 2 + √3 i / 2 and u = - 1 / 2 - √3 i / 2

    u = ( - 1 + √3 i ) / 2 and u = - ( 1 + √3 i ) / 2

    This means that this equation has not only real solutions, there are also complex conjugate solutions.

    Substitute back u = x², and solve for x.

    x² = - 1 / 2 + √3 i / 2

    x² = - ( 1 - √3 i ) / 2 = [ - ( 1 + √3 i ) / 2 ]

    x = ± √ [ - ( 1 + √3 i ) / 2 ]²

    x = - ( 1 + √3 i ) / 2 and ( 1 + √3 i ) / 2

    x² = - ( 1 / 2 + √3 i / 2 )

    - ( 1 + √3 i ) / 2 = - ( 1 - √3 i / 2 )²

    x = ± √ [ - ( 1 - √3 i ) / 2 ]²

    x = - ( 1 - √3 i ) / 2 and - [ - ( 1 - √3 i ) / 2 ]

    x = ( - 1 + √3 i ) / 2 and ( 1 - √3 i / ) / 2

    Finally the solutions are:

    x = - 1 , x = 1 , x = - ( 1 + √3 i ) / 2 , x = ( 1 + √3 i ) / 2 , x = ( - 1 + √3 i ) / 2 , x = ( 1 - √3 i ) / 2

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    Hi everyone, my name is Emerson. I was wondering if anybody could help me on a test. I'm stuck on some, but I think I know some others. Thank you! 1. Translate the phrase "nine more than two times a number" in to an algebraic

    asked by Emerson on December 7, 2015
  2. Mathematics

    using ruler and a pair of compasses only construct a quadrilateral ABCD with ab equal 9 cm BC equal 10 cm equal 7.5 cm and ab is equal to 5 degree and less than b a d equal 135 degree what type of quadrilateral is ABCD

    asked by Angel on June 15, 2020
  3. algebra

    graph the following function on the set of axes below. f(x) = lxl, -3 less than or equal to x less than 1 and 4, 1 less than or equal to x less than or equal to 8

    asked by amy on March 31, 2016
  4. Algebra 2 -Linear Programming

    Find the values of x and y that maximize or minimize the objective function. x+y < or equal to 8 2x+y < or equal to 10 x> or equal to 0, y > or equal to 0 A. (0,5) Maximum value is 100 B.(1,7) Maximum value is 220 C.(2,6) Maximum

    asked by LaH on October 26, 2011
  5. Math

    please check my work. 1. The number of words (x) Tina can type per minute is at Least 50. write and graph and inequality. (I broke it down to these 2 choices because the A & D was incorrect) B. x is less than or equal to 50 C. x

    asked by kenziemack on November 10, 2017
  1. STATISTIC

    A two-tailed test is conducted at the 0.10 significance level. What is the P-value required to reject the null hypothesis? A. Greater than or equal to .010 B. Greater than or equal to 0.05 C. Less than or equal to 0.10 D. Less

    asked by Vedrana on February 25, 2016
  2. Probability

    Let X be a random variable, and let ck=E[Xk]. We assume that X is such that ck exists (and is finite) for all k∈{0,1,2,⋯}. Define Yk=Xk. Please express your answers in terms of X,c1,c2,…,ck,… (for c1,c2,... please enter

    asked by Anonymous on May 8, 2018
  3. Physics

    For an angle measured from the positive x-axis, the values of sin and cos are always which of the following? A) less than -1 B) greater than or equal to -1 and less than or equal to 1 C) less than or equal to -1 or greater than or

    asked by Tamela harris on September 7, 2019
  4. Math 222

    Please help me solve this system of equations? Sorry addition/equal key is not working. -2x plus 9y equal 3 -3x-4y equal 22

    asked by Tika on April 27, 2015
  5. ALGEBRA 1 HELP

    The description below represents Function A and the table represents Function B: Function A The function is 2 more than 5 times x. Function B x y -1 2 0 5 1 8 Which statement is correct about the slope and intercept of the two

    asked by Tony on August 1, 2017
  6. math

    7. The tennis team is selling key chains as a fundraiser. If its goal is to raise is at least 180, how many key chains must it sell at $2.25 each to meet that goal? Write and solve an inequality. a. 2.25k (greater than or equal

    asked by john on October 18, 2016

You can view more similar questions or ask a new question.