Surds, Log and Indices
An interesting question showing these 3 topics can be tested in a single question
Question
Express
as ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vag-qYVrTuccu8lp5ipFNuPbEBL9oMB3AImqYEAM95kl-WGqa-wRe66oxPeuUI_wzBkbygjHzFPp_XMq13Qc3QowiURV4S9k9ChGofNHS6tPOIUHxsloM9EnFjo6hpWl58Fal0w7ufPtqY-JZVnSgT=s0-d)
i) Hence show that![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_snRsm17vXaSdoQ72PFMwo0z4syFPBifbWZnDu6Ucdg2izHZuqu8pzRobLRjC8oukRaXPzQwUVlpFiEIf3NxhN80_ePfrnzk6IY1Kxdz1tpY9CCLNMXoD_24BgfKUsEkn_yRYP9g3Rnwnp-1SOAjbx0qgzbNcGEA5h4VJp3tkbPz1-XpFgM5IMrwK5L0_flxj8bPZ2WI6N-xPiwfYWUaz5p=s0-d)
ii) Given that
, prove that ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vEMrcEw3EtrAf9-hsZWvR7Kn157hG1FPs1Mudp_P0WBIZuwxayHqRWUARSTXpqhyftYwbJuz474d7NpJBwQdxcjquzilM7u6nMVN0EsEX1kEPi22N8X5QsowJB9cz8fji47HodXTJzoHdTSHtMlkt9OvKlxEqxY-XqUuaHzlh2BVjZBY8=s0-d)
Techniques tested
Answer
Technique used: Rationalization
i)
Techniques used:
Indices:![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uRG_b7Zu1DXGLunok_IiCNg1KopVz5vXUzaUOx937--mSck39jFZ2CQ7HQ5XefG0XaXn0naDWlH2pZgnZiym1FwPTklqYTb5eBrgOziVgRU_PNPhYhLrPgGGe1RVL-JOmubT4CZWeaQvbaIeDcvrC1Mr3eivHmwmUPHY2_GB_tz26CryESpwc=s0-d)
Power Law:![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sDy3VEvM1nPbkRTIrSCEKQ0ROINDM37A0BUfXrIxKlNEanqTgQeaxq_KOu6a9Bm9bGhuX9XynT4B7p401zOaBI284jBNGocsV3dPmFyu_oKNkwsTSVn07crm3oDMFfG_HMqa9ngkTRFxR6P4MTFgO8cuJcbA=s0-d)
From LHS
ii)
This part of the question requires us to use the Substitution technique
![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ui3bz4AXseTZ6aXdNQP_4tBXZwmqRwYSa7GQjWQb9oOlAjnsM-hEevSdTdwmLnrBFaxv5xr5O1XLb8Tl-kD1QpC_a2p8YXZBMVoa_XKO19-i3FDGCx1fao_VBQ85QSEEHlGu8Fhknba_C9HGrN2sBQism8KFZFyHoAAoNLmjQlKueDVDrBqdvi-EUkGG74cFc8nu7YmM5uN0Nr2k0s_RQnt9VxD9G1ZIG37RuuHq6Pe6LHRFqaTEaWTZSW66_oAbm-_JGnSTINJ5hThI35IvCgVsSUw6ofM7j8OmVZ03jrsZcHc7iakjOXSEwZEB87Q98buY-om6iDmsnkgcs6e9CGYU3wyMw9txMHoLM9JTU3YpFaN8vQ6JhQr4i0EO84S7KoEcrqfxIlGxhxX1RkEkF-7EYTfgNcqYFwiLyBYpbg2LUzs2sy8yJibPGIdK5KqLYK-0MYBBUxKUA1DFwdjxrbMtFQ6TR22cBXX8VBgMibdOCNl8K8DYk=s0-d)
Using the quadratic formula,
Recall that![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_soI4bennifGCJbjeH1Z9KBvDEH4Of2ok2mgJem2S2UfH4ikE305cdeR3n24WuIpsGYHejh25RXRLsYcT2Irjg3epj-RVEHecyKDLEOS73mPGZWIQk6H0elm-Ko-jsOhky2TvXh=s0-d)
therefore![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_she2NBSXGn6CYUZWU6puXdc98TC39EMXWHuvFEi4SseEv7PzFIMt-2rJOK3Q2KfrPszccgohSKFrPrApfkjC2uEkDEW0pUvD-IaGz_itwaN4JtBXXtuJL_iW7qLZtWFsLZDx0AqdeNt6xSs0tQiNJYqXkRTTef6qzh=s0-d)
Technique used: Taking log on both sides
![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vXvR1XFhs-b5zY3ke7m5BMzhgjv_xXB84ZuNL6apyo9SoRTNY_QqmokhBoeVamHgTIgcA-cjDbGLCeCIhmsh_-3U3kc3o96Enj6mNff6ZMnD8aPyMBNsYvMKwzHhY8seC3-52M4riIn9wNKb97EDfRR0H8aAYSZdqMY_w9bt4df7hqafshA_oaSSJuBnp0cIFKVJ0VDf7siJ8QIqDtNknk3Dtt-HHvwP1timY8tG-6Dw1Sul_n3exhA-l-lMaClEJkVWh-WFrMP7q8qN1GLUdzgMkoH3tTO3sQk3hISvj2Ph_MlYbl-8bu3uXcHuBwbh_Q8fQCecwy6o0mEUQAv1hywLu2Xef4UV8Wa7i7CYQb_oTd=s0-d)
![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uZVzPuCvZWTHW5LRbmqVTWGQ1FR6thPdJJ2BZmeqadkRc-RcIKi7zZ7ZGRT_h0nVp5sutaYJbj8xrcKX_iff3xAj0CIkkaUzDeHjd187XCQfumdRHc-J-7lgvbpRmELv7Xoo9NXG0cusIyDr-jSZFp4hzg6iszbLMo841i0kmHia4kMHiTyNAfsqLPI9SCH0aXzFfL0Y_axP4HPwhU4FiDFjqFRU4FjkDaIw-Co4s5o_YBoSJdH1BUVV3t93lq3O5uDj7xx1gEuxinM1hQxAKWczdh_uCzxtOb-rVcZXLCCDw2clcerYieho7QxwNSOHo0Z8gOEFJwAXMFPPMg86i9pOaB6qwryUVpoHgZ9-0IPTeDdQ=s0-d)
Recall that in part (i) we showed that![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_snRsm17vXaSdoQ72PFMwo0z4syFPBifbWZnDu6Ucdg2izHZuqu8pzRobLRjC8oukRaXPzQwUVlpFiEIf3NxhN80_ePfrnzk6IY1Kxdz1tpY9CCLNMXoD_24BgfKUsEkn_yRYP9g3Rnwnp-1SOAjbx0qgzbNcGEA5h4VJp3tkbPz1-XpFgM5IMrwK5L0_flxj8bPZ2WI6N-xPiwfYWUaz5p=s0-d)
Interesting question, dont you think ?
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For a revision of surds, click here
For a revision of logarithms, click here
For a revision of indices, click here
Question
Express
i) Hence show that
ii) Given that
Techniques tested
- Rationalization
- Indices:
- Power Law:
Answer
Technique used: Rationalization
i)
Techniques used:
Indices:
Power Law:
From LHS
ii)
This part of the question requires us to use the Substitution technique
Using the quadratic formula,
Recall that
therefore
Technique used: Taking log on both sides
Recall that in part (i) we showed that
Interesting question, dont you think ?
For more interesting questions subscribe to my blog by entering your email address at the top right hand corner
Have an interesting question to share? send me an email at yapye at (@) yahoo dot com dot sg
For a revision of surds, click here
For a revision of logarithms, click here
For a revision of indices, click here
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