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Part 1. Pick a two-digit number greater than 25. Rewrite your two-digit number as a difference of two numbers. Show how to use the identity (x − y)2 = x2 − 2xy + y2 to square your number without using a calculator.Part 2. Choose two values, a and b, each between 8 and 15. Show how to use the identity a3 + b3 = (a + b)(a2 − ab + b2) to calculate the sum of the cubes of your numbers without using a calculator.

Answer :

Solution:-1

Number be 27

[tex]\\ \sf\longmapsto 27^2[/tex]

[tex]\\ \sf\longmapsto (30-3)^2[/tex]

[tex]\\ \sf\longmapsto 30^2-2(30)(3)+3^2[/tex]

[tex]\\ \sf\longmapsto 900-180+9[/tex]

[tex]\\ \sf\longmapsto 909-180[/tex]

[tex]\\ \sf\longmapsto 729[/tex]

[tex]\\ \sf\longmapsto 27^2[/tex]

Hence verified

Solution:-2

  • a be 10
  • b be 9

[tex]\\ \sf\longmapsto 10^3+9^3[/tex]

[tex]\\ \sf\longmapsto (10+9)(10^2-10\times 9+9^2)[/tex]

[tex]\\ \sf\longmapsto 19(100-90+81)[/tex]

[tex]\\ \sf\longmapsto 19\times 91[/tex]

[tex]\\ \sf\longmapsto 1729[/tex]

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