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.
1. Answer :
(a - b)3
We know that
(a + b)3 = 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)3 = a3 - 3ab(a - b) - b3
2. Answer :
(y + 3)3
We know that
(a + b)3 = 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)3 = 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)3 = 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)3 = 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)3 = 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)3 = a3 + 3a2b + 3ab2 + b3
Substitute a = 100, and b = -2.
(100 - 2)3 = 1003 + 3(100)2(-2) + 3(100)(-2)2 + (-2)3
983 = 1000000 + 3(10000)(-2) + 3(100)(4) - 8
= 1000000 - 60000 + 1200 - 8
= 941192
8. Answer :
993 = (100 - 1)3
We know that
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Substitute a = 100, and b = -1.
(100 - 1)3 = 1003 + 3(100)2(-1) + 3(100)(-1)2 + (-1)3
993 = 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)3 = 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)3 = 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 + b3 = (a + b)3 - 3ab(a + b)
Substitute a = 3x, and b = 4y.
(3x)3 + (4y)3 = (3x + 4y)3 - 3(3x)(4y)(3x + 4y)
27x3 + 64y3 = (3x + 4y)3 - 36(xy)(3x + 4y)
Substitute 3x + 4y = 10 and xy = 2.
27x3 + 64y3 = (10)3 - 36(2)(10)
= 1000 - 720
= 280
12. Answer :
We know that
a3 - b3 = (a - b)3 + 3ab(a - b)
Substitute a = x, and b = y.
x3 - y3 = (x - y)3 + 3xy(x - y)
Substitute x - y = 5 and xy = 14.
x3 - y3 = 53 + 3(14)(5)
= 125 + 210
= 335
13. Answer :
We know that
a3 + b3 = (a + b)3 - 3ab(a + b)
Substitute a = a, and b = 1/a.
a3 + (1/a)3 = (a + 1/a)3 - 3a(1/a)(a + 1/a)
a3 + 1/a3 = (a + 1/a)3 - 3(a + 1/a)
Substitute a + 1/a = 6.
a3 + 1/a3 = (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 - b3 = (a - b)3 + 3ab(a - b)
Substitute a = y, and b = 1/y.
y3 - (1/y)3 = (y - 1/y)3 + 3y(1/y)(y - 1/y)
y3 - 1/y3 = (y - 1/y)3 + 3(y - 1/y)
Substitute (y - 1/y)3 = 27 and y - 1/y = 3.
y3 - 1/y3 = 27 + 3(3)
= 27 + 9
= 36
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