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

1. ## (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.

$45x^6\, =\, 9x^4$

$\dfrac{45x^6}{\color{blue}{x^4}}\, =\, \dfrac{9x^4}{\color{blue}{x^4}}$

$45x^2\, =\, 9$

$\dfrac{45x^2}{\color{blue}{45}}\, =\, \dfrac{9}{\color{blue}{45}}$

$x^2\, =\, \dfrac{1}{5}$

$\color{blue}{\sqrt{\strut \color{black}{x^2}\,}}\, =\, \color{blue}{\sqrt{\strut \color{black}{\frac{1}{5}}\,}}$

$x\, =\, \color{blue}{\pm}\dfrac{1}{\sqrt{\strut 5\,}}$

$\color{red}{ x\, =\, \pm\, \dfrac{\sqrt{\strut 5\,}}{5} }$

Thanks in advance! (Black and blue is my working out. Red is what I missed according to the answers page)

2. 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)

$\dfrac{5\sqrt{3}}{\sqrt{7}} \; \cdot \;$$\dfrac{\sqrt{7}}{\sqrt{7}}$$= \dfrac{5\sqrt{21}}{7}$

The denominator has been rationalized.

3. Originally Posted by Tofi
(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.

$45x^6 = 9x^4$

$45x^6 - 9x^4 = 0$

$9x^4(5x^2 - 1) = 0$

From the quartic factor set equal to zero, that gives x = 0.

4. Originally Posted by mmm4444bot
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)

$\dfrac{5\sqrt{3}}{\sqrt{7}} \; \cdot \;$$\dfrac{\sqrt{7}}{\sqrt{7}}$$= \dfrac{5\sqrt{21}}{7}$

The denominator has been rationalized.
Thanks!

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