Question

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Answer 1

Q,1 Find the cubes of :

7, -14, 2 2/5 , 0.6 .

(Answers is the attachment)

Q.2 Find the ones digit in cube of

43, 98 , 251 ,312,205

Answer:

1.43 cube = 79507 so the one digit number is 7.

2. 98 cube = 951,192 so the one digit number is 2 .

3. 251 cube = 15813251 so the one digit number is 1.

4. 312 cube = 30371328 so the one digit number is 8.

5. 205 cube = 8615125 so the one digit number is 5

Note that : Whenever we find cube of any number the cube of it unit is known as one digit number.

Q.3 Find the value of :

a) 6³ - 5³

= 6³-5³ = 1 +(6×5×3)

= 6³ - 5³ = 1 + 90 = 91

b) 11³ - 10³

11³ - 10³ = 1 +(11×10×3)

11³ - 10³ = 1 +(330) = 331

Q.4 Check whether the following numbers are perfect cube or not ?

1] 648

2] 1296

Answer :

1] It is not a perfect cube root because,

648 = 2×2×2×3×3×3×3 .

A cube root is found when the prime factors have 3 of each number.

Hence, 3×3×3 must be multiplied with 648 to get a perfect cube root.

Hence, it is not a perfect cube

2] In the attachment

Answer 2

Questions

1. Find the cubes of ⇒ 7, -14, 2²/₅, 0.6

2. Find the ones digit in the cube of ⇒ 43, 98, 251, 312, 205

3. Observe the patter given below ⇒

2³ - 1³ = 1 + (2 × 1 × 3)

3³ - 2³ = 1 + (3 × 2 × 3)

4³ - 3³ = 1 + (4 × 3 × 3)

Now find the value of (a) 6³ - 5³ (b) 11³ - 10³

4. Check whether the following numbers are perfect cubes or not?⇒ 648, 1296

$\rule{300}{1}$

Answers

1. A cube of a number is same as multiplying the number 3 times to itself. So let's find the cube of the following.

(i) 7³ = 7 × 7 × 7 = 343

∴ Cube of 7 is 343

(ii) (-14)³ = -14 × -14 × -14 = -2744

∴ Cube of -14 is -2744

(iii) (2²/₅)³ = (¹²/₅)³ = ¹²/₅ × ¹²/₅ × ¹²/₅ = ¹⁷²⁸/₁₂₅

∴ Cube of 2²/₅ is ¹⁷²⁸/₁₂₅

(iv) (0.6)³ = 0.6 × 0.6 × 0.6 = 0.216

∴Cube of 0.6 is 0.216

$\rule{300}{1}$

2. To find the units digit of the cube of the following number we need to cube the units place and whatever number we get it's unit digit would be the units digit of the cue of the following number. So let's find the unit's digit of the cube.

(i) 3³ = 3 × 3 × 3 = 81

∴ The units digit of the cube of 43 would be 1.

(ii) 8³ = 8 × 8 × 8 = 512

∴ The units digit of the cube of 98 would be 2.

(iii) 1³ = 1 × 1 × 1 = 1

∴ The units digit of the cube of 251 would be 1.

(iv) 2³ = 2 × 2 × 2 = 8

∴ The units digit of the cube of 312 would be 8.

(v) 5³ = 5 × 5 × 5 = 125

∴ The units digit of the cube of 205 would be 5.

$\rule{300}{1}$

3. Using the pattern given we shall solve the given.

(i) 6³ - 5³ = 1 + (6 × 5 × 3) = 1 + 90 = 91

(ii) 11³ - 10³ = 1 + (11 × 10 × 3) = 1 + 330 = 331

$\rule{300}{1}$

4. To see if these are perfect cubes we can prime factorize them and if the product of primes form groups in which 3 numbers are there without leaving a prime number it is a perfect cube.

(i) 648

After prime factorization we get this ⇒ 2 × 2 × 2 × 3 × 3 × 3 × 3

When we group them with 3 in each group we get ⇒ (2×2×2)×(3×3×3)×3

∴ 3 is left.

∴ 648 is not a perfect cube.

(ii) 1296

After prime factorization we get this ⇒ 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3

When we group them with 3 in each group we get ⇒ (2×2×2)×2×(3×3×3)×3

∴ 2 and 3 are left.

∴ 1296 is not a perfect cube.

$\rule{300}{1}$