Algebra

Surds

Examples

To reduce ³2808. Decompose the number into its prime factors, thus, ³2808=³2³⋅3³⋅13=6³13

Examples

³𝑎¹⁵𝑏¹⁰𝑐⁸=𝑎¹⁵³𝑏¹⁰³𝑐⁸³=𝑎⁵𝑏⁹³⁺¹³𝑐⁶³⁺²³=𝑎⁵𝑏³𝑏¹³𝑐²𝑐²³=𝑎⁵𝑏³𝑐²⋅³𝑏𝑐²

Examples

To bring 5⁴3 to an entire surd. 5⁴3=⁴5⁴3=5⁴1875

Examples

𝑥²³𝑦¹⁵𝑧¹²=𝑥²⁰³⁰𝑦⁶³⁰𝑧¹⁵³⁰=³⁰𝑥²⁰𝑦⁶𝑧¹⁵

Examples

To rationalise fractions having surds in their demoninators. 17=77 1³7=³49³7⋅³49=³49³7×49=³497 39−80=3(9+80)(9−80)(9+80)=3(9+80)81−80=3(9+80)

Examples

11+23−2=1+23+2(1+23−2)(1+23+2)=1+23+2(1+23)2−2=1+23+211+43  =(1+23+2)(11−43)(11+43)(11−43)=(1+23+2)(11−43)121−48  =(1+23+2)(11−43)73

Examples

1³3−2=13¹³−2¹² Put 3¹³=𝑥, 2¹²=𝑦, and take 6 the L.C.M. of the denominators 2 and 3, then 1𝑥−𝑦=𝑥⁵+𝑥⁴𝑦+𝑥³𝑦²+𝑥²𝑦³+𝑥𝑦⁴+𝑦⁵𝑥⁶−𝑦⁶ therefore 1313−212=313⋅5+313⋅4212+313⋅3212⋅2+313⋅2212⋅3+313212⋅4+212⋅5313⋅6−212⋅6 133−2=339+33322+6+23922+433+429−8  =339+3672+6+26648+433+42 Similarly, 1³3+2 Here the result will be the same as in the last example if the signs of the even terms be changed.

Properties

A surd cannot be partly rational; that is, 𝑎 cannot be equal to 𝑏±𝑐. Proved by squaring. The product of two unlike squares is irrational; 7×3=21, an irrational quantity. The sum or difference of two unlike surds cannot produce a single surd; that is, 𝑎+𝑏 cannot be equal to 𝑐. By squaring. If 𝑎+𝑚=𝑏+𝑛; then 𝑎=𝑏 and 𝑚=𝑛. Above Theorems are proved indirectly. if 𝑎+𝑏=𝑥+𝑦 then 𝑎−𝑏=𝑥−𝑦 by squaring and above theorem.

Examples

To express in two terms 7+26 Let 7+26=𝑥+𝑦 then by squaring and about theorem 𝑥+𝑦=7 and by about theorem 𝑥−𝑦=7²−(2 6)²=49−24=5 ∴ 𝑥=6 and 𝑦=1 Result 6+1

General Formula

General formula for the same 𝑎±𝑏=½(𝑎+𝑎²−𝑏)±½(𝑎−𝑎²−𝑏) Observe that no simplification is effected unless 𝑎²−𝑏 is a perfect square.

Examples

To simplify ³𝑎+𝑏 Assume ³𝑎+𝑏=𝑥+𝑦 Let 𝑐=³𝑎²−𝑏. Then 𝑥 must be found by trial from the cubic equation 4𝑥³−3𝑐𝑥=𝑎 and 𝑦=𝑥²−𝑐 No simplification is effected unless 𝑎²−𝑏 is a perfect cube.

Examples

³7+52=𝑥+𝑦 𝑐=³49−50=−1 4𝑥³+3𝑥=7; ∴ 𝑥=1, 𝑦=2 Result 1+2

Examples

³93−113=𝑥+𝑦 two different surds Cubing, 93−113=𝑥𝑥+3𝑥𝑦+3𝑦𝑥+𝑦𝑦; ∴ 93=(𝑥+3𝑦)𝑥112=(3𝑥+𝑦)𝑦}; ∴ 𝑥=3 and 𝑦=2

Examples

To simplify (12+43+45+215) Assume (12+43+45+215)=𝑥+𝑦+𝑧 Square, and equate corresponding surds. Result 3+4+5

Theorems

To express ^𝑛𝐴±𝐵 in the form of two surds, where 𝐴 and 𝐵 are one or both quadratic surds and 𝑛 is odd. Take 𝑞 such that 𝑞(𝐴²−𝐵²) may be a perfect 𝑛^th power, say 𝑝^𝑛. Take 𝑠 and 𝑡 the nearest integers to ^𝑛𝑞(𝐴+𝐵)² and ^𝑛𝑞(𝐴−𝐵)², then ^𝑛𝐴+𝐵=12 ^2𝑛𝑞{𝑠+𝑡+2𝑝±𝑠+𝑡−2𝑝}

Examples

To reduce ⁵893+1092 Here 𝐴=893, 𝐵=1092 𝐴²−𝐵²=1; ∴ 𝑝=1 and 𝑞=1. ⁵𝑞(𝐴+𝐵)²=9+𝑓⁵𝑞(𝐴−𝐵)²=1−𝑓}, 𝑓 being a proper fractio; ∴ 𝑠=9, 𝑡=1. Result, 12(9+1+2±9+1−2)=3+2

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive