Class 8 Maths Squares and Square Roots Notes

Squares and Square Roots – Quick Notes

Square of a Number

• Definition: The square of a number $n$n is the number multiplied by itself.

$$n^2 = n \times n$$n2=n×n

• Examples:

$$5^2 = 25, \quad (-4)^2 = 16$$52=25,(−4)2=16

• Properties of Squares:

1. Square of a positive number = positive
2. Square of a negative number = positive
3. Square of 0 = 0
4. The square of an integer always ends with certain digits (0, 1, 4, 5, 6, 9)

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Square Root of a Number

• Definition: The square root of a number $x$x is a number $y$y such that $y^2 = x$y2=x.

$$\sqrt{x} = y \implies y^2 = x$$x​=y⟹y2=x

• Examples:
  $\sqrt{25} = 5, \quad \sqrt{16} = 4$25​=5,16​=4
• Properties:

1. $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$a×b​=a​×b​
2. $\sqrt{a / b} = \sqrt{a} / \sqrt{b}$a/b​=a​/b​

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Methods to Find Square Roots

1. Prime Factorization Method
   • Example: Find $\sqrt{144}$144​
   • Factorization: $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3$144=2×2×2×2×3×3
   • Pair factors: $(2 \cdot 2)(2 \cdot 2)(3 \cdot 3) = 12$(2⋅2)(2⋅2)(3⋅3)=12
   • So, $\sqrt{144} = 12$144​=12
2. Long Division Method
   • Divide the number into pairs of digits from right to left.
   • Find the largest number whose square ≤ the first pair.
   • Subtract, bring down the next pair, double the quotient, find the next digit.
3. Using Estimation
   • Useful for non-perfect squares (like $\sqrt{50} \approx 7.07$50​≈7.07)

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Applications of Squares and Square Roots

• Area of a square: $\text{Area} = (\text{side})^2$Area=(side)2
• Solving quadratic equations
• Real-life problems in geometry, algebra, and mensuration

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Squares and Square Roots – MCQ Q&A

1. Q: Square of 7?
   A: $7^2 = 49$72=49
2. Q: Square of -8?
   A: $(-8)^2 = 64$(−8)2=64
3. Q: $\sqrt{81} = ?$81​=?
   A: 9
4. Q: Which of the following is a perfect square? 20, 25, 30
   A: 25 ✅
5. Q: $\sqrt{144} = ?$144​=?
   A: 12
6. Q: Square of 0?
   A: 0
7. Q: Product property of square roots: $\sqrt{16 \times 25} = ?$16×25​=?
   A: $\sqrt{16} \times \sqrt{25} = 4 \times 5 = 20$16​×25​=4×5=20
8. Q: Quotient property: $\sqrt{36/9} = ?$36/9​=?
   A: $\sqrt{36}/\sqrt{9} = 6/3 = 2$36​/9​=6/3=2
9. Q: Find $\sqrt{196}$196​ using prime factorization
   A: $196 = 2^2 \cdot 7^2 \implies \sqrt{196} = 2 \cdot 7 = 14$196=22⋅72⟹196​=2⋅7=14
10. Q: Which of these numbers is not a perfect square? 64, 81, 90
    A: 90 ❌
11. Q: Long division method is used for?
    A: Finding square roots of large numbers
12. Q: Estimate $\sqrt{50}$50​
    A: $\approx 7.07$≈7.07
13. Q: Area of a square with side 9 cm?
    A: $9^2 = 81\ cm^2$92=81 cm2
14. Q: Square root of 1?
    A: 1
15. Q: Square root of 0?
    A: 0
16. Q: Square of 15?
    A: $15^2 = 225$152=225
17. Q: Square root of 225?
    A: 15
18. Q: Can the square of a number be negative?
    A: No ✅
19. Q: Find $\sqrt{121}$121​
    A: 11
20. Q: Square of -0.5?
    A: $(-0.5)^2 = 0.25$(−0.5)2=0.25