Class 8 Maths – Cubes and Cube Roots Notes

Cubes and Cube Roots – Quick Notes (Class 8 NCERT)

Cube of a Number

• Definition: The cube of a number $n$n is the number multiplied by itself three times:

$$n^3 = n \times n \times n$$n3=n×n×n

• Examples:

$$2^3 = 8, \quad (-3)^3 = -27$$23=8,(−3)3=−27

• Properties of Cubes:

1. Cube of a positive number = positive
2. Cube of a negative number = negative
3. Cube of 0 = 0
4. Cubes grow faster than squares

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Cube Root of a Number

• Definition: The cube root of a number $x$x is a number $y$y such that $y^3 = x$y3=x:

$$\sqrt[3]{x} = y \implies y^3 = x$$3x​=y⟹y3=x

• Examples:
  $\sqrt[3]{8} = 2, \quad \sqrt[3]{-27} = -3$38​=2,3−27​=−3
• Properties:

1. $\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$3a⋅b​=3a​⋅3b​
2. $\sqrt[3]{a / b} = \sqrt[3]{a} / \sqrt[3]{b}$3a/b​=3a​/3b​

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Methods to Find Cube Roots

1. Prime Factorization Method
   • Example: Find $\sqrt[3]{216}$3216​
   • Factorization: $216 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 = 2^3 \cdot 3^3$216=2⋅2⋅2⋅3⋅3⋅3=23⋅33
   • Take one from each triple: $2 \cdot 3 = 6$2⋅3=6
   • So, $\sqrt[3]{216} = 6$3216​=6
2. Estimation Method
   • For non-perfect cubes: Find the nearest cubes and estimate
   • Example: $\sqrt[3]{50} \approx 3.684$350​≈3.684 (between $3^3 = 27$33=27 and $4^3 = 64$43=64)

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Applications of Cubes and Cube Roots

• Volume of a cube: $V = (\text{side})^3$V=(side)3
• Real-life problems in geometry and algebra
• Solving cubic equations in higher mathematics

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Cubes and Cube Roots – MCQ Q&A

1. Q: Cube of 3?
   A: $3^3 = 27$33=27
2. Q: Cube of -2?
   A: $(-2)^3 = -8$(−2)3=−8
3. Q: Cube root of 125?
   A: $\sqrt[3]{125} = 5$3125​=5
4. Q: Cube of 0?
   A: 0
5. Q: Cube of 1?
   A: 1
6. Q: $(-5)^3 = ?$(−5)3=?
   A: -125
7. Q: Cube root of -64?
   A: -4
8. Q: Product property: $\sqrt[3]{8 \cdot 27} = ?$38⋅27​=?
   A: $\sqrt[3]{8} \cdot \sqrt[3]{27} = 2 \cdot 3 = 6$38​⋅327​=2⋅3=6
9. Q: Quotient property: $\sqrt[3]{125 / 8} = ?$3125/8​=?
   A: $\sqrt[3]{125}/\sqrt[3]{8} = 5/2$3125​/38​=5/2
10. Q: Find $\sqrt[3]{343}$3343​
    A: 7
11. Q: Is cube of a negative number positive or negative?
    A: Negative ✅
12. Q: Volume of a cube with side 4 cm?
    A: $4^3 = 64\ cm^3$43=64 cm3
13. Q: Cube root of 1?
    A: 1
14. Q: Cube root of 0?
    A: 0
15. Q: Find $\sqrt[3]{216}$3216​ using prime factorization
    A: $2^3 \cdot 3^3 \implies 2 \cdot 3 = 6$23⋅33⟹2⋅3=6
16. Q: Which of these is a perfect cube? 8, 12, 18
    A: 8 ✅
17. Q: Estimate $\sqrt[3]{50}$350​
    A: $\approx 3.684$≈3.684
18. Q: Cube root of -27?
    A: -3
19. Q: Can the cube of a number be zero?
    A: Yes, if the number is 0 ✅
20. Q: Cube of 10?
    A: $10^3 = 1000$103=1000