Algebra, Algebraic Techniques, Roots / Radicals

This topic deals with roots of numbers (square roots, cube roots, etc.), denoted by radical signs (`\sqrt{}`, `\sqrt[n]{}`). Questions involve simplifying radical expressions, performing operations with radicals, rationalizing denominators, and solving equations containing radicals.

• Question

  Which number is larger: `sqrt(2016+2017)` or `sqrt(2016)+sqrt(2017)`?

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Inequalities Algebra -> Algebraic Techniques -> Roots / Radicals

• Question

  Find the value of the expression without a calculator:

  `(sqrt(sqrt(2)-1))/(sqrt(sqrt(2)+1))+ (sqrt(sqrt(3)-sqrt(2)))/(sqrt(sqrt(3)+sqrt(2)))+ (sqrt(2-sqrt(3)))/(sqrt(2+sqrt(3)))`

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  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Algebraic Techniques -> Roots / Radicals Algebra -> Algebraic Techniques -> Multiplication by the Conjugate

• Question

  Given distinct rational numbers a, b, c, prove that `sqrt{1/(a-b)^2+1/(b-c)^2 +1 /(c-a)^2}`

  is rational.

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  Topics:

  Number Theory Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 3

• Question

  Given natural numbers m, n such that `m/n <= sqrt 23`, prove that `m/n+3/{mn} <= sqrt 23`

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Algebra -> Inequalities Proof and Example -> Proof by Contradiction Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 4

• The Matchstick Root

  Move one matchstick to make a correct equality:

  

  Topics:

  Algebra -> Algebraic Techniques -> Roots / Radicals Puzzles and Rebuses -> Matchstick Puzzles

• Volume of a Rectangular Prism

  Given a rectangular prism whose face areas are 6, 10, and 15. Find its volume.

  

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  Topics:

  Geometry -> Solid Geometry / Geometry in Space Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 5-6 Question 4

• Quadratic Equations

  Solve the equations:

  a: `({x^2+6}/{x^2-4})^2=({5x}/{4-x^2})^2`

  b: `7(x+1/x)-2(x^2+1/x^2)=9`

  c: `sqrt{x+2sqrt{x-1}}-sqrt{x-2sqrt{x-1}}=2`

  Topics:

  Algebra -> Equations Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Junior Science Team Exercises, 2024-2025, Assignment 1 Question 1

• Partitioned Roots

  Find the sum `1/{sqrt1+sqrt2}+1/{sqrt2+sqrt3}+...+1/{sqrt99+sqrt100}`.

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  Topics:

  Algebra -> Algebraic Techniques -> Telescoping Sums Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Junior Science Team Exercises, 2024-2025, Assignment 1 Question 2