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Lesson 4 of 4

A Square and A Cube · Lesson 4 of 4

Chapter Summary and Practise

“Review squares, cubes, roots, patterns and prime-factor methods through mixed practice.”

Learning Objectives

• Review the main properties of square and cubic numbers. • Distinguish perfect squares from perfect cubes using patterns and prime factorisation. • Find square roots and cube roots of suitable whole numbers. • Use odd-number patterns, nearby perfect powers and factor grouping to solve problems. • Apply the chapter ideas to mixed reasoning questions.

SUMMARY

A square number is obtained by multiplying a number by itself, while a cube is obtained by multiplying a number by itself three times. Perfect squares connect naturally with areas of squares; perfect cubes connect with the number of unit cubes in a larger cube. Square root and cube root reverse these operations.

IdeaSquaresCubes
Repeated multiplicationn² = n × nn³ = n × n × n
Prime-factor structureEqual factors group in pairsEqual factors group in triplets
Inverse operationSquare root √Cube root ∛
Odd-number patternFirst n odd numbers sum to n²Special groups of consecutive odd numbers sum to cubes
Trailing zerosNumber of zeros is doubledNumber of zeros is tripled
Squares use pairs; cubes use triplets Perfect square 144 = 2 × 2 × 2 × 2 × 3 × 3 (2 × 2) (2 × 2) (3 × 3) √144 = 12 Perfect cube 216 = 2 × 2 × 2 × 3 × 3 × 3 (2 × 2 × 2) (3 × 3 × 3) ∛216 = 6
Prime factor grouping for squares and cubes

Units digits are useful clues. A perfect square can end only in 0, 1, 4, 5, 6 or 9, so endings 2, 3, 7 and 8 immediately rule out a square. Perfect cubes can end in any units digit, so cube questions require different clues.

Worked Example: Consecutive Squares

Problem
Given 125² = 15625, find 126² without long multiplication.

  1. 1.The difference between 125² and 126² is the next odd number.
  2. 2.That odd number is 2 × 126 − 1 = 251.
  3. 3.126² = 15625 + 251 = 15876.
Worked Example: Making a Perfect Square

Problem
Find the smallest number by which 9408 must be multiplied to make a perfect square. Then find the square root of the product.

  1. 1.9408 = 2⁶ × 3 × 7².
  2. 2.For a perfect square, every prime exponent must be even.
  3. 3.Only the factor 3 is unpaired, so multiply by 3.
  4. 4.9408 × 3 = 2⁶ × 3² × 7².
  5. 5.Its square root is 2³ × 3 × 7 = 168.
Worked Example: Estimating a Square Root

Problem
Estimate √250 using nearby perfect squares.

  1. 1.15² = 225 and 16² = 256.
  2. 2.Therefore, 15 < √250 < 16.
  3. 3.250 is much closer to 256 than to 225.
  4. 4.So √250 is close to 16, but slightly less than 16.
Worked Example: Cube Root

Problem
Find ∛10648.

  1. 1.22³ = 22 × 22 × 22.
  2. 2.22 × 22 = 484, and 484 × 22 = 10648.
  3. 3.Therefore, ∛10648 = 22.
Worked Example: Making a Perfect Cube

Problem
What is the smallest number by which 1323 must be multiplied to make a perfect cube?

  1. 1.1323 = 3³ × 7².
  2. 2.For a perfect cube, every prime exponent must be a multiple of 3.
  3. 3.The exponent of 7 needs one more factor of 7.
  4. 4.Multiply by 7: 1323 × 7 = 3³ × 7³ = (3 × 7)³.
  5. 5.So the product is 21³, and its cube root is 21.

Quiz

Quick check

Which statement correctly compares prime factors of perfect squares and perfect cubes?

Quick check

Which number can immediately be ruled out as a perfect square by its last digit?

Quick check

If 24² = 576, what is √576?

Quick check

Which number is a perfect cube?

Quick check

Which expression equals 4³?

Practice Problems

Practice Problems
  1. Which of 2032, 2048, 1027 and 1089 are not perfect squares? Give a reason for each conclusion.
  2. Find the length of the side of a square whose area is 441 m².
  3. Find the smallest perfect square that is divisible by 4, 9 and 10.
  4. Find the cube roots of 27000 and 10648.
  5. Decide whether each statement is true or false and explain: (a) the cube of every odd number is even; (b) no perfect cube ends in 8; (c) a perfect square can have an odd number of trailing zeros; (d) prime factors of a perfect cube can be grouped into triplets.

Key Takeaways

Key Takeaways

• A square is formed by multiplying a number by itself; a cube uses three equal factors. • Perfect squares can be recognised through several clues, but a units-digit clue alone cannot always prove that a number is a square. • Consecutive square numbers differ by consecutive odd numbers. • Square roots and cube roots reverse squaring and cubing. • Prime factors of a perfect square group into pairs; prime factors of a perfect cube group into triplets. • Nearby perfect squares help estimate square roots that are not whole numbers. • Pattern recognition, factorisation and reasoning are as important as calculation in this chapter.

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A Pinch of History

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