CUBE OF A BINOMIAL WORKSHEET

Problem 1 :

Expand (a - b)3.

Problem 2 :

Expand (y + 3)3.

Problem 3 :

Expand (x - 5)3.

Problem 4 :

Expand (5x + 3y)3.

Problem 5 :

Expand (3p - 4q)3.

Problem 6 :

Expand (x + 1/y)3.

Problem 7 :

Evaluate using identity : 983.

Problem 8 :

Evaluate using identity : 993.

Problem 9 :

Evaluate using identity : 1013.

Problem 10 :

Evaluate using identity : 10013.

Problem 11 :

Find 27x3 + 64y3, if 3x + 4y = 10 and xy = 2.

Problem 12 :

Find x3 - y3, if x - y = 5 and xy = 14.

Problem 13 :

If a + 1/a = 6, then find the value of a3 + 1/a3.

Problem 14 :

If (y - 1/y)3 = 27, then find the value of y3 - 1/y3.

Answers

1. Answer :

(a - b)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Replace 'b' by '-b'.

(a + (-b))3 = a3 + 3a2(-b) + 3a(-b)2 + (-b)3

(a - b)3 = a3 - 3a2b + 3ab2 - b3

or

(a - b)= a3 - 3ab(a - b) - b3

2. Answer :

(y + 3)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = y, and b = 3.

(y + 3)3 = y3 + 3y2(3) + 3y(3)2 + 33

= y3 + 9y2 + 3y(9) + 27

= y3 + 9y2 + 27y + 27

3. Answer :

(x - 5)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = x, and b = -5.

(x - 5)3 = x3 + 3x2(-5) + 3x(-5)2 + (-5)3

= x3 - 15x2 + 3x(25) - 125

= x3 - 15x2 + 75x - 125

4. Answer :

(5x + 3y)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = 5x, and b = 3y.

(5x + 3y)3 = (5x)3 + 3(5x)2(3y) + 3(5x)(3y)2 + (3y)3

= 125x3 + 3(25x2)(3y) + 3(5x)(9y2) + 27y3

= 125x3 + 225x2y + 135xy2 + 27y3

5. Answer :

(3p - 4q)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = 3p, and b = -4q.

(3p - 4q)3 = (3p)3 + 3(3p)2(-4q) + 3(3p)(-4q)2 + (-4q)3

= 27p3 + 3(9p2)(-4q) + 3(3p)(16q2) + (-64q3)

= 27p3 - 108p2q + 144pq2 - 64q3

6. Answer :

(x + 1/y)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = x, and b = 1/y.

(x + 1/y)3 = x3 + 3(x)2(1/y) + 3(x)(1/y)2 + (1/y)3

= x3 + 3x2/y + 3x/y2 + 1/y3

7. Answer :

983 = (100 - 2)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = 100, and b = -2.

(100 - 2)3 = 1003 + 3(100)2(-2) + 3(100)(-2)2 + (-2)3

98= 1000000 + 3(10000)(-2) + 3(100)(4) - 8

= 1000000 - 60000 + 1200 - 8

= 941192

8. Answer :

993 = (100 - 1)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = 100, and b = -1.

(100 - 1)3 = 1003 + 3(100)2(-1) + 3(100)(-1)2 + (-1)3

99= 10,00,000 + 3(10000)(-1) + 3(100)(1) - 1

= 10,00,000 - 30,000 + 300 - 1

= 970,299

9. Answer :

1013 = (100 + 1)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = 100, and b = 1.

(100 + 1)3 = 1003 + 3(100)2(1) + 3(100)(1)2 + 13

101 = 10,00,000 + 3(10000)(1) + 300 + 1

= 10,00,000 + 30,000 + 300 + 1

= 10,30,301

10. Answer :

10013 = (1000 + 1)3

We know that

(a + b)= a3 + 3a2b + 3ab2 + b3

Substitute a = 1000, and b = 1.

(1000 + 1)3 = 10003 + 3(1000)2(1) + 3(1000)(1)2 + (1)3

(1001)3 = 10003 + 3(1000)2(1) + 3(1000)(1)2 + (1)3

= 10,00,000,000 + 3(1,000,000)(1) + 3(1000)(1) + 1

= 10,00,000,000 + 3,000,000 + 3000 + 1

= 10,03,003,001

11. Answer :

We know that

a3 + b= (a + b)3 - 3ab(a + b)

Substitute a = 3x, and b = 4y.

(3x)3 + (4y)= (3x + 4y)3 - 3(3x)(4y)(3x + 4y)

27x3 + 64y= (3x + 4y)3 - 36(xy)(3x + 4y)

Substitute 3x + 4y = 10 and xy = 2.

27x3 + 64y= (10)3 - 36(2)(10)

= 1000 - 720

= 280

12. Answer :

We know that

a3 - b= (a - b)3 + 3ab(a - b)

Substitute a = x, and b = y.

x3 - y= (x - y)3 + 3xy(x - y)

Substitute x - y = 5 and xy = 14.

x3 - y= 53 + 3(14)(5)

= 125 + 210

= 335

13. Answer :

We know that

a3 + b= (a + b)3 - 3ab(a + b)

Substitute a = a, and b = 1/a.

a3 + (1/a)= (a + 1/a)3 - 3a(1/a)(a + 1/a)

a3 + 1/a= (a + 1/a)3 - 3(a + 1/a)

Substitute a + 1/a = 6.

a3 + 1/a= (6)3 + 3(6)

= 216 + 18

= 198

14. Answer :

(y - 1/y)3 = 27

(y - 1/y)3 = 33

y - 1/y = 3

We know that

a3 - b= (a - b)3 + 3ab(a - b)

Substitute a = y, and b = 1/y.

y3 - (1/y)= (y - 1/y)3 + 3y(1/y)(y - 1/y)

y3 - 1/y= (y - 1/y)3 + 3(y - 1/y)

Substitute (y - 1/y)3 = 27 and y - 1/y = 3.

y3 - 1/y= 27 + 3(3)

= 27 + 9

= 36

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