A Square and A Cube

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A square number is obtained by multiplying a number by itself. It is written as n² = n × n. First five square numbers: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25

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They are called squares because they represent the area of a square. For example, a square with side length 3 units has area 3 × 3 = 9 square units. The area equals the square of the side length.

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Perfect squares can only end with the digits: 0, 1, 4, 5, 6, or 9. Perfect squares CANNOT end with: 2, 3, 7, or 8. This property helps identify non-squares quickly.

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(1000)² = 1,000,000 (has 6 zeros) Rule: If a number has n zeros at the end, its square will have 2n zeros at the end. Since 1000 has 3 zeros, (1000)² has 2 × 3 = 6 zeros.

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1 + 3 + 5 + 7 + 9 = 25 = 5² General Rule: The sum of the first n odd numbers equals n². Here we have 5 odd numbers, so the sum is 5² = 25.

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The nth odd number = 2n - 1 Examples: • 1st odd number: 2(1) - 1 = 1 • 5th odd number: 2(5) - 1 = 9 • 10th odd number: 2(10) - 1 = 19

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36² = 35² + 36th odd number 36th odd number = 2(36) - 1 = 71 Therefore: 36² = 1225 + 71 = 1296

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Square root: If y = x², then x is the square root of y. For 64: Since 8² = 64 and (-8)² = 64 The square roots of 64 are +8 and -8 Written as: √64 = ±8 (We usually consider only the positive square