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Thread: (Help) Simplifying by factoring question for 45x^6 = 9x^4

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    Question (Help) Simplifying by factoring question for 45x^6 = 9x^4

    Hi, so I've been self-teaching myself maths as a hobby for a little while now. And I came across this problem that's confused me a bit. I thought I followed all the necessary steps to simplify it, but my answer book told me I didn't take it that one step further. I'm not sure how it got this final answer and my book frustratingly hasn't taught me how it came to this conclusion.



    [tex]45x^6\, =\, 9x^4[/tex]

    [tex]\dfrac{45x^6}{\color{blue}{x^4}}\, =\, \dfrac{9x^4}{\color{blue}{x^4}}[/tex]

    [tex]45x^2\, =\, 9[/tex]

    [tex]\dfrac{45x^2}{\color{blue}{45}}\, =\, \dfrac{9}{\color{blue}{45}}[/tex]

    [tex]x^2\, =\, \dfrac{1}{5}[/tex]

    [tex]\color{blue}{\sqrt{\strut \color{black}{x^2}\,}}\, =\, \color{blue}{\sqrt{\strut \color{black}{\frac{1}{5}}\,}}[/tex]

    [tex]x\, =\, \color{blue}{\pm}\dfrac{1}{\sqrt{\strut 5\,}}[/tex]

    [tex]\color{red}{ x\, =\, \pm\, \dfrac{\sqrt{\strut 5\,}}{5} }[/tex]



    Thanks in advance! (Black and blue is my working out. Red is what I missed according to the answers page)
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    Last edited by stapel; 01-31-2018 at 04:30 PM. Reason: Typing out the text in the graphic; creating useful subject line.

  2. #2
    Elite Member mmm4444bot's Avatar
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    Hi. Your result and the red result are two different forms of the same answer.

    Were you instructed to always "rationalize the denominator", by chance?

    To rationalize a denominator means to multiply both the numerator and denominator by the same radical expression, to get a ratio with no radical in the denominator.

    For example, rationalize the denominator in 5√(3/7)

    [tex]\dfrac{5\sqrt{3}}{\sqrt{7}} \; \cdot \; [/tex][tex] \dfrac{\sqrt{7}}{\sqrt{7}} [/tex][tex] = \dfrac{5\sqrt{21}}{7}[/tex]

    The denominator has been rationalized.
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

  3. #3
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    Quote Originally Posted by Tofi View Post
    (Black and blue is my working out. Red is what I missed according to the answers page)


    The answers page is incorrect if you presented it fully. In the context of this specific problem,
    it is not allowable to divide by the variable, because information is lost.

    [tex]45x^6 = 9x^4[/tex]

    [tex]45x^6 - 9x^4 = 0[/tex]

    [tex]9x^4(5x^2 - 1) = 0[/tex]

    From the quartic factor set equal to zero, that gives x = 0.
    Last edited by lookagain; 01-30-2018 at 02:28 PM.

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    Quote Originally Posted by mmm4444bot View Post
    Hi. Your result and the red result are two different forms of the same answer.

    Were you instructed to always "rationalize the denominator", by chance?

    To rationalize a denominator means to multiply both the numerator and denominator by the same radical expression, to get a ratio with no radical in the denominator.

    For example, rationalize the denominator in 5√(3/7)

    [tex]\dfrac{5\sqrt{3}}{\sqrt{7}} \; \cdot \; [/tex][tex] \dfrac{\sqrt{7}}{\sqrt{7}} [/tex][tex] = \dfrac{5\sqrt{21}}{7}[/tex]

    The denominator has been rationalized.
    Thanks!

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