Introduction
When we multiply a number by itself, we get its square. When we multiply a number by itself three times, we get its cube. Understanding squares and cubes helps us calculate areas, volumes, and solve many real-life problems.
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Key Concepts
Square of a number: If n is any whole number, then n x n = n2 is called the square of n. For example, 52 = 25.
Perfect square (or square number): A natural number that is the square of another natural number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Properties of perfect squares:
- A perfect square never ends in 2, 3, 7, or 8.
- A perfect square always ends in 0, 1, 4, 5, 6, or 9.
- Squares of even numbers are even; squares of odd numbers are odd.
- The number of zeros at the end of a perfect square is always even.
Cube of a number: n x n x n = n3. For example, 43 = 64.
Perfect cube: A number that is the cube of a natural number. Examples: 1, 8, 27, 64, 125, 216.
- Key Formulas:
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- (a + b)(a - b) = a2 - b2
- Sum of first n odd numbers = n2
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Worked Examples
Find the square of 23.
Step 1: Write 23 = 20 + 3.
Step 2: (20 + 3)2 = 202 + 2 x 20 x 3 + 32 = 400 + 120 + 9 = 529.
Is 196 a perfect square? Find its square root.
Step 1: Prime factorize 196: 196 = 2 x 2 x 7 x 7 = (2 x 7)2 = 142.
Step 2: Each prime factor appears in pairs, so yes, 196 is a perfect square. Square root = 14.
Find the smallest number by which 75 must be multiplied to make it a perfect square.
Step 1: Prime factorize 75: 75 = 3 x 5 x 5.
Step 2: Factor 3 is unpaired. Multiply by 3 to get 225 = 32 x 52 = 152.
Answer: Multiply by 3.
Find the cube of 12.
123 = 12 x 12 x 12 = 144 x 12 = 1728.
Is 512 a perfect cube?
Prime factorize 512: 512 = 29 = (23)3 = 83. Yes, 512 is a perfect cube; its cube root is 8.
Find the smallest number to divide 500 to make it a perfect cube.
500 = 22 x 53. Factor 2 appears twice (not a group of 3). Divide by 22 = 4. Then 500/4 = 125 = 53.
Using the pattern of consecutive odd numbers, verify that 42 = 13 + 15 (sum of consecutive odds).
12 = 1; 22 = 1+3 = 4; 32 = 1+3+5 = 9; 42 = 1+3+5+7 = 16. Correct!
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Common mistakes
- Confusing the square root with half the number. Remember: square root of 16 is 4, not 8.
- Thinking a number ending in 1 or 9 is always a perfect square — it must also pass prime factorization test.
- Forgetting to check that ALL prime factors appear an even number of times for perfect squares (or multiple of 3 for cubes).
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Summary
Squares and cubes are foundational concepts in number theory. A perfect square has all prime factors in pairs; a perfect cube has all prime factors in groups of three. The identities (a+b)2 and (a-b)2 speed up mental calculation. The sum of first n odd numbers always equals n2.