Algebra Frenzy - Part 1

Algebra Level pending

What does m + 1 m + 2 m + 1 m \dfrac{m+\frac{1}{m}+2}{\sqrt{m}+\frac{1}{\sqrt{m}}} equal to?

m + 1 m \displaystyle\sqrt{m}+\frac{1}{\sqrt{m}} m 2 + 1 m 2 \displaystyle m^2+\frac{1}{m^2} m + 1 m \displaystyle m+\frac{1}{m} m \displaystyle m

Alice Smith by Alice Smith , 20, China

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2 solutions

Chew-Seong Cheong
May 18, 2019

m + 2 + 1 m m + 1 m = ( m + 1 m ) 2 m + 1 m = m + 1 m \dfrac {m+2+\frac 1m}{\sqrt m + \frac 1{\sqrt m}} = \dfrac {\left(\sqrt m + \frac 1{\sqrt m}\right)^2}{\sqrt m + \frac 1{\sqrt m}} = \boxed{\sqrt m + \dfrac 1{\sqrt m}}

2 Helpful 0 Interesting 1 Brilliant 0 Confused
Chris Lewis
May 18, 2019

Rationalising the denominator and cancelling factors works, but it's quicker to notice that ( m + 1 m ) 2 = m + 2 + 1 m \left(\sqrt m+\frac{1}{\sqrt m}\right)^2=m+2+\frac{1}{m}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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