Question Number 9123 by j.masanja06@gmail.com last updated on 20/Nov/16
$$\mathrm{simplify} \\ $$$$\left(\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}+\mathrm{1}\right)^{−\mathrm{1}/\mathrm{2}} −\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} \right)/\mathrm{x}^{\mathrm{2}} \\ $$
Commented by tawakalitu last updated on 20/Nov/16
![((x^2 (x + 1)^(−1/2) − (x + 1)^(1/2) )/x^2 ) = [x^2 . (1/((x + 1)^(1/2) )) − (((x + 1)^(1/2) )/1)] ÷ x^2 = [(x^2 /((x + 1)^(1/2) )) − (((x + 1)^(1/2) )/1)] × (1/x^2 ) Find the LCM in the braket. = [((x^2 − (x + 1))/((x + 1)^(1/2) ))] × (1/x^2 ) = ((x^2 − x − 1)/(x^2 (√(x + 1))))](https://www.tinkutara.com/question/Q9124.png)
$$\frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}\:+\:\mathrm{1}\right)^{−\mathrm{1}/\mathrm{2}} \:−\:\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} } \\ $$$$=\:\left[\mathrm{x}^{\mathrm{2}} .\:\frac{\mathrm{1}}{\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }\:−\:\frac{\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }{\mathrm{1}}\right]\:\boldsymbol{\div}\:\mathrm{x}^{\mathrm{2}} \\ $$$$=\:\left[\frac{\mathrm{x}^{\mathrm{2}} }{\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }\:−\:\frac{\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }{\mathrm{1}}\right]\:×\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{LCM}\:\mathrm{in}\:\mathrm{the}\:\mathrm{braket}. \\ $$$$=\:\left[\frac{\mathrm{x}^{\mathrm{2}} \:−\:\left(\mathrm{x}\:+\:\mathrm{1}\right)}{\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }\right]\:×\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$=\:\frac{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{x}\:−\:\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{x}\:+\:\mathrm{1}}} \\ $$