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CBSE Class 8 · Mathematics

A Square and A Cube

Official NCERT chapter from Ganita Prakash Part I (book code hegp1). ExamMaster notes are original teaching at CBSE Class 8 depth.

This lesson follows the official chapter “A Square and A Cube” in Ganita Prakash Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 8
  • Easy level
  • 8 concepts

1Square Numbers

A square number is a whole number times itself: 8 × 8 = 64, so 64 is square. You can picture it as an 8-by-8 grid. 65 is not a square — there is no whole n with n × n = 65.

A number that looks square in a drawing of a window is not this test.

Figure. A square number is a count that fills an n-by-n array. 1, 4 and 9 are 1², 2² and 3² — tiles that make a square, not a name pinned on any four-sided picture.

How it works

  1. Take a whole number n8.
  2. Multiply it by itself64.
  3. Call 64 a square number; refuse 65No whole grid.

Is 81 square?

Is 81 a square number? If yes, of which n?

  • Try 9 × 981
  • Try 8 × 864 too small
  • Yes9² = 81

Pro tip. A square number is n × n for a whole n.

2Cubic Numbers

A cube number is n × n × n: 4 × 4 × 4 = 64, so 64 is also a cube. Picture a 4-by-4-by-4 block. 27 is 3³; 8 is 2³. 16 is square but not a cube.

Calling every “boxy” number a cube is a look, not n×n×n.

Figure. A cube number is n × n × n. The bars are 1, 8 and 27 — 1³, 2³ and 3³. Cubes grow fast: 3³ is already more than three times 2³. No isometric box is drawn, because that primitive is not here.

How it works

  1. Take n4.
  2. Multiply three copies4 × 4 × 4 = 64.
  3. Call 64 a cube; check 16 is only a squareTwo different tests.

3A Pinch of History

People have named square and cube numbers for a long time; the names come from the grid and the block. History here is a pinch — it is not a proof that 64 is 8².

A famous name does not replace n × n.

Figure. A pinch of history: odd numbers build squares. 1 makes 1², add 3 to make 2², add 5 to make 3². The newest L-shaped layer is the gnomon. 1 + 3 + 5 = 9.

How it works

  1. Keep the pinch as a noteThe names are old.
  2. Return to n × n or n × n × nThis number.
  3. Check by multiplyingHistory is not the product.

4A definition is a test you can run

“Square number” means you can name n and show n × n. If you cannot run that product, you have a label only.

Memorising a list without a product is not the definition.

Figure. A definition is a test you can run. Nine tiles make a 3-by-3 square, so 9 is a square number. Eight tiles make a 2-by-4 rectangle, so 8 fails. Four sides is not the test.

How it works

  1. Propose a number36.
  2. Find n with n × n equal to it6.
  3. If no whole n exists, it is not squareThe test.

5Name the given before the unknown

The given is the number (or n). The unknown is whether it is square/cube, or the value of n² or n³. Write the given before you hunt a factor.

Assuming 100 is 8² because 100 looks round is a silent given.

Figure. Name the given before the unknown. The given is the side 5. The unknown is the square count. Only after the side is named do we write 5² = 25.

How it works

  1. Copy the given number or nThe given.
  2. Name square, cube, or the productThe unknown.
  3. Then multiply or searchGiven first.

6One worked case is enough at this class

One square and one cube are enough. A page of 1², 2², 3² clones does not add a new product-rule.

Ten twins of 64 are still one idea if you already showed 8² and 4³.

Figure. One worked case is enough at this class: a 4-by-4 square holds 16 unit tiles, so 4² = 16. The same test will run on any other whole-number side.

How it works

  1. Compute one n² and one n³The case.
  2. A second only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A tiled floor photo is a setting. The maths is n × n. Counting tiles in a photo that is not a square grid and calling it 8² is the swap-miss.

A pretty courtyard is not 64 unless you counted 8 by 8.

Figure. The miss is swapping the school name for the picture. A square has four equal sides. A four-sided rectangle that is longer than it is tall is not a square number's picture, and calling it one is the miss.

How it works

  1. Keep the floor as a storyA setting.
  2. Write n × nThe maths.
  3. ComputeDo not swap.

8Check by the opposite action or the opposite test

Check a square-claim by taking square root sense: 7² = 49, 8² = 64, so 50 is not square. Check a cube by n×n×n, not by “it looks 3-D”.

Saying “64 looks square” without n is not the opposite test.

Figure. Check by the opposite action. Squaring 5 gives 25; the square root of 25 returns 5. If the return trip fails, the square claim was wrong.

How it works

  1. State n² or n³The claim.
  2. Multiply again or sandwich between two known squaresThe check.
  3. Mend if no whole n fitsHonest.
64 is
  1. Both 8² and 4³ — run each product
  2. Only a square
  3. Not a cube

Two tests; both pass for 64.

Notes

  • The official chapter title is “A Square and A Cube”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • n² = n × n
  • n³ = n × n × n

Recap

Hold these pegs from the official chapter “A Square and A Cube”. The wording is ExamMaster’s teaching, not a textbook recap.

Square Numbers
A square number is a whole number times itself: 8 × 8 = 64, so 64 is square.
Cubic Numbers
A cube number is n × n × n: 4 × 4 × 4 = 64, so 64 is also a cube.
A Pinch of History
People have named square and cube numbers for a long time; the names come from the grid and the block.
A definition is a test you can run
“Square number” means you can name n and show n × n.
Name the given before the unknown
The given is the number (or n). The unknown is whether it is square/cube, or the value of n² or n³. Write the given before you hunt a factor.
One worked case is enough at this class
One square and one cube are enough. A page of 1², 2², 3² clones does not add a new product-rule.

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