Squares and Cubes – Complete Notes

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1. Square of a Number

• Definition: The square of a number $n$ is $n \times n$, written as $n^2$.

• Read as: “n squared”

• Example: ${\mathrm{5²}}^{}=25$

• Inverse Operation: Square root ( $\sqrt{\text{\hspace{0.25em}\hspace{0.05em}√}}$).

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2. Perfect Squares

• Definition: A number whose square root is a whole number.

• List: $1,4,9,16,25,36,49,64,81,100,\dots$

• Properties:

  1. If a number ends in 2, 3, 7, or 8 → Not a perfect square.

  2. Perfect squares end in 0, 1, 4, 5, 6, or 9.

  3. Perfect squares can only have an even number of zeros at the end.

  4. Square of even numbers → even; square of odd numbers → odd.

  5. A number ending in an odd number of zeros is never a perfect square.

  6. Difference between squares of two consecutive numbers:

     $$(n+1)^2 - n^2 = 2n + 1$$

  7. Pythagorean Triplet: $m^2 + n^2 = p^2$. Example: $3^2 + 4^2 = 5^2$.

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3. Patterns in Perfect Squares

• Last digit patterns:

  • Ends in 1 or 9 → Square ends in 1

  • Ends in 4 or 6 → Square ends in 6

  • Ends in 5 → Square ends in 5

• Sum of first $n$ odd numbers:

  $$1 + 3 + 5 + \dots + (2n - 1) = n^2$$

  Example: $1 + 3 + 5 = 9 = 3^2$

• Numbers between two consecutive perfect squares:

  $$2m \quad \text{where } m \text{ and } m+1 \text{ are consecutive numbers}$$

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4. Square Root

• Definition: If $y = x^2$, then $x$ is the square root of $y$.

• Notation: $\sqrt{y}$

• Example: $\sqrt{49} = \pm 7$ (positive and negative roots for integers).

• Perfect Square Test: A number is a perfect square if prime factors can be paired into two identical groups.

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5. Cube of a Number

• Definition: The cube of a number $n$ is $n \times n \times n$, written as $n^3$.

• Example: $3^3 = 27$

• Can be positive or negative: $(-2)^3 = -8$.

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6. Perfect Cubes

• Definition: A number whose cube root is a whole number.

• Example: $\sqrt[3]{27} = 3$ → 27 is a perfect cube.

• List: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \dots$

• Properties:

  1. Cube of even number → even; cube of odd number → odd.

  2. Cube of a number ending in:

     • 0 → ends in 0

     • 1 → ends in 1

     • 2 → ends in 8

     • 3 → ends in 7

     • 4 → ends in 4

     • 5 → ends in 5

     • 6 → ends in 6

     • 7 → ends in 3

     • 8 → ends in 2

     • 9 → ends in 9

  3. Sum of cubes of first $n$ natural numbers:

     $$1^3 + 2^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2$$

  4. In prime factorisation, if each prime factor occurs three times, the number is a perfect cube.

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7. Cube Root

• Definition: If $y = x^3$, then $x$ is the cube root of $y$.

• Notation: $\sqrt[3]{y}$

• Example: $\sqrt[3]{-8} = -2$.

• Perfect Cube Test: Prime factors can be split into three identical groups.

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8. Special Numbers – Hardy–Ramanujan (Taxicab) Numbers

• Definition: Numbers that can be expressed as the sum of two cubes in two different ways.

• Example:

  $$1729 = 1^3 + 12^3 = 9^3 + 10^3$$

• First few taxicab numbers: 1729, 4104, 13832, ...

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9. Key Patterns to Remember

Concept	Pattern / Formula
Difference between consecutive squares	$2n+1$
Sum of first $n$ odd numbers	$n^2$
Sum of first $n$ cubes	$\left( \frac{n(n+1)}{2} \right)^2$
Last digit pattern for cubes	(0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9)