Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Find the cube of each of the following binomial expression
(a) (1/x + y/3)
(b) (3/x − 2/x2)
(c) (2x + 3/x)
(d) (4 − 1/3x)
(a) Given,
(1/x + y/3))3
The above equation is in the form of (a + b)3 = a3 + b3 + 3ab(a + b)
We know that, a = 1/x, b = y/3
By using (a + b)3 formula
= (1/x)3 + (y/3)3 + 3(1/x)( y/3)(1/x + y/3)
= 1/x3 + y3/27 + 3 * 1/x * y/3(1/x + y/3)
= 1/x3 + y3/27 + y/x(1/x + y/3)
= 1/x3 + y3/27 + (y/x * 1/x) +(y/x * y3)
= 1/x3 + y3/27 + y/x2 + y2/3x)
Hence,
(1/x + y/3))3 = 1/x3 + y3/27 + y/x2 + y2/3x)
(b) Given,
((3/x−2/x2))3
The above equation is in the form of (a - b)3 = a3 - b3 - 3ab(a - b)
We know that, a = 3/x, b = 2/x2
By using (a - b)3 formula
((3/x − 2/x2))3 = (3x)3 - (2/x2)3 - 3(3/x)( 2/x2)(3/x - 2/x2)
= 27/x3 - 8/x6 - 3 * 3/x * 2/x2(3/x - 2/x2)
= 27/x3 - 8/x6 - 18/x3(3/x - 2/x2)
= 27/x3 - 8/x6 - (18/x3 * 3/x) + (18/x3 * 2/x2)
= 27/x3 - 8/x6 - 54/x4 + 36/x5
Hence, ((3/x−2/x2))3 = 27/x3 - 8/x6 - 54/x4 + 36/x5
(c) Given,
(2x + 3/x)3
We know that, a = 2x, b = 3/x
= 8x3 + 27/x3 + 18x/x(2/x + 3/x)
= 8x3 + 27/x3 + 18x/x(2x + 3/x)
= 8x3 + 27/x3 + (18 * 2x) + (18 * 3/x)
= 8x3 + 27/x3 + 36 × 54/x)
The cube of (2x + 3/x)3 = 8x3 + 27/x3 + 36 × 54/x)
(d) Given,
(4 − 1/3x)3
We know that, a = 4, b = 1/3x
(4 − 1/3x)3 = 43 - (1/3x)3 - 3(4)(1/3x)(4 - 1/3x)
= 64 - 1/27x3 - 12/3x(4 - 1/3x)
= 64 - 1/27x3 - 4/x(4 - 1/3x)
= 64 - 1/27x3 - (4/3x * 4) + (4/3x * 1/3x)
= 64 - 1/27x3 - 16/x + (4/3x2)
The cube of (4 − 1/3x)3 = 64 - 1/27x3 - 16/x + (4/3x2)
Simplify each of the following
(a) (x + 3)3 + (x - 3)3
(b) (x/2 + y/3)3 - (x/2 - y/3)3
(c) (x + 2/x)3 + (x − 2/x)3
(d) (2x - 5y)3 - (2x + 5y)3
(a) (x + 3)3 + (x – 3)3
The above equation is in the form of a3 + b3 = (a + b)(a2 + b2 – ab)
We know that, a = (x + 3), b = (x – 3)
By using (a3 + b3) formula
= (x + 3 + x – 3)[(x + 3)3 + (x – 3)3 – (x + 3)(x – 3)]
= 2x[(x2 + 32 + 2*x*3) + (x2 + 32 – 2*x*3) – (x2 – 32)]
= 2x[(x2 + 9 + 6x) + (x2 + 9 – 6x) – x2 + 9]
= 2x[(x2 + 9 + 6x + x2 – 9 – 6x – x2 + 9)]
= 2x(x2 + 27)
= 2x3 + 54x
Hence, the result of (x + 3)3 + (x – 3)3 is 2x3 + 54x
(b) (x/2 + y/3)3 - (x/2 – y/3)3
The above equation is in the form of a3 - b3 = (a - b)(a2 + b2 + ab)
We know that, a = (x/2 + y/3)3, b = (x/2 - y/3)3
By using (a3 - b3) formula
= [((x/2 + y/3)3 - ((x/2 − y/3)3)][((x/2 + y/3)3)2((x/2 - y/3)3)2 - ((x/2 + y/3)3)((x/2 - y/3)3)
= (x/3 + y/3 − x/2 + y/3)[((x/2)2 + (y/3)2 + (2xy/6)) + ((x/2)2 + (y/3)2 − (2xy/6)) + ((x/2)2 − (y/3)2)]
= 2y/3[(x2/4 + y2/9 + 2xy/6) + (x2/4 + y2/9 − 2xy/6) + x2/4 − y2/9]
= 2y/3[x2/4 + y2/9 + 2xy/6 + x2/4 + y2/9 − 2xy/6 + x2/4 − y2/9]
= 2y/3[x2/4 + y2/9 + x2/4 + x2/4]
= 2y/3[3x2/4 + y2/9]
= x2y/2 + 2y3/27
Hence, the result of (x/2 + y/3)3 - (x/2 - y/3)3 = x2y/2 + 2y3/27
The above equation is in the form of a3 + b3 = (a + b)(a2 + b2 - ab)
We know that, a = (x + 2x)3, b = (x − 2x)3
= (x + 2/x + x − 2/x)[(x + 2/x)2 + (x − 2/x)2 − ((x + 2/x)(x − 2/x))]
= (2x)[(x2 + 4/x2 + 4x/x) + (x2 + 4/x2 − 4x/x) − (x2 − 4/x2)
= (2x)[(x2 + 4/x2 + 4x/x + x2 + 4/x2 − 4x/x − x2 + 4/x2)
= (2x)[(x2 + 4/x2 + 4/x2 + 4/x2)
= (2x)[(x2 + 12/x2)
= 2x3 + 24/x
Hence, the result of (x + 2/x)3 + (x − 2/x)3 = (2x)[(x2 + 12/x2)
Given, (2x - 5y)3 - (2x + 5y)3
We know that, a = (2x - 5y), b = (2x + 5y)
= (2x – 5y – 2x – 5y)[(2x – 5y)2 + (2x + 5y)2 + ((2x – 5y) * (2x + 5y))]
= (-10y)[(4x2 + 25y2 – 20xy) + (4x2 + 25y2 + 20xy) + 4x2 – 25y2]
= (-10y)[ 4x2 + 25y2 – 20xy + 4x2 + 25y2 + 20xy + 4x2 – 25y2]
= (-10y)[4x2 + 4x2 + 4x2 + 25y2]
= (-10y)[12x2 + 25y2}
= -120x2y – 250y3
Hence, the result of (2x – 5y)3 – (2x + 5y)3 = -120x2y – 250y3
If a + b = 10 and ab = 21, Find the value of a3 + b3
Given,
a + b = 10, ab = 21
we know that, (a + b)3 = a3 + b3 + 3ab(a + b) ... 1
substitute a + b = 10 , ab = 21 in eq 1
⟹ (10)3 = a3 + b3 + 3(21)(10)
⟹ 1000 = a3 + b3 + 630
⟹ 1000 – 630 = a3 + b3
⟹ 370 = a3 + b3
Hence, the value of a3 + b3 = 370
If a - b = 4 and ab = 21, Find the value of a3 - b3
a - b = 4, ab = 21
we know that, (a - b)3 = a3 - b3 - 3ab(a - b) -------- 1
substitute a - b = 4 , ab = 21 in eq 1
⟹ (4)3 = a3 - b3 - 3(21)(4)
⟹ 64 = a3 - b3 - 252
⟹ 64 + 252 = a3 - b3
⟹ 316 = a3 - b3
Hence, the value of a3 - b3 = 316
If (x + 1/x) = 5, Find the value of x3 + 1/x3
Given, (x + 1/x) = 5
We know that, (a + b)3 = a3 + b3 + 3ab(a + b) ... 1
Substitute (x + 1/x) = 5 in eq1
(x + 1/x)3 = x3 + 1/x3 + 3(x * 1/x)(x + 1/x)
53 = x3 + 1/x3 + 3(x * 1/x)(x + 1/x)
125 = x3 + 1/x3 + 3(x + 1/x)
125 = x3 + 1/x3 + 3(5)
125 = x3 + 1/x3 + 15
125 – 15 = x3 + 1/x3
x3 +1/x3 = 110
Hence, the result is x3 + 1/x3 = 110
If (x − 1/x) = 7, Find the value of x3 − 1/x3
Given, If (x − 1/x) = 7
We know that, (a - b)3 = a3 - b3 - 3ab(a - b) ... 1
Substitute (x − 1/x) = 7 in eq 1
(x − 1/x)3 = x3 − 1/x3 - 3(x * 1/x)(x - 1/x)
73 = x3 - 1/x3 - 3(x - 1/x)
343 = x3 -1/x3 - (3 * 7)
343 = x3 - 1/x3 - 21
343 + 21 = x3 - 1/x3
x3 - 1/x3 = 364
hence, the result is x3 - 1/x3 = 364
If (x − 1/x) = 5, Find the value of x3 − 1/x3
Given, If (x − 1/x) = 5
Substitute (x − 1/x) = 5 in eq 1
(x − 1/x)3 = x3 - 1/x3 - 3(x * 1/x)(x - 1/x)
53 = x3 - 1/x3 - 3(x - 1/x)
125 = x3 - 1/x3 - (3 * 5)
125 = x3 - 1/x3 - 15
125 + 15 = x3 - 1/x3
x3 - 1/x3 = 140
Hence, the result is x3 - 1/x3 = 140
If (x2 + 1/x2) = 51, Find the value of x3 − 1/x3
Given, (x2 + 1/x2) = 51
We know that, (x - y)2 = x2 + y2 - 2xy .... 1
Substitute (x2 + 1/x2) = 51 in eq 1
(x - 1/x)2 = x2 + 1/x2 - 2 * x * 1/x
(x - 1/x)2 = x2 + 1/x2 - 2
(x - 1/x)2 = 51 - 2
(x - 1/x)2 = 49
(x – 1/x) =√49
(x - 1/x) = ±7
We need to find x3 − 1/x3
So, a3 - b3 = (a - b)(a2 + b2 + ab)
x3 − 1/x3 = (x - 1/x)(x2 + 1/x2 + (x * 1/x)
We know that,
(x -1/x) = 7 and (x2 + 1/x2) = 51
x3 − 1/x3 = 7(51 + 1)
x3 − 1/x3 = 7(52)
x3 − 1/x3 = 364
Hence, the value of x3 − 1/x3 = 364
If (x2 + 1/x2) = 98, Find the value of x3 + 1/x3
Given, (x2 + 1/x2) = 98
We know that, (x + y)2 = x2 + y2 + 2xy ... 1
Substitute (x2 + 1/x2) = 98 in eq 1
(x + 1/x)2 = x2 + 1/x2 + 2 * x * 1/x
(x + 1/x)2 = x2 + 1/x2 + 2
(x + 1/x)2 = 98 + 2
(x + 1/x)2 = 100
(x + 1/x) = √100
(x + 1/x) = ± 10
We need to find x3 + 1/x3
So, a3 + b3 = (a + b)(a2 + b2 – ab)
x3 + 1/x3 = (x + 1/x)(x2 + 1/x2 - (x * 1/x)
(x + 1/x) = 10 and (x2 + 1/x2) = 98
x3 + 1/x3 = 10(98 - 1)
x3 + 1/x3 = 10(97)
x3 + 1/x3 = 970
Hence, the value of x3 + 1/x3 = 970
If 2x + 3y = 13 and xy = 6, Find the value of 8x3 + 27y3
Given, 2x + 3y = 13, xy = 6
(2x + 3y)3 = 132
⟹ 8x3 + 27y3 + 3(2x)(3y)(2x + 3y) = 2197
⟹ 8x3 + 27y3 + 18xy(2x + 3y) = 2197
Substitute 2x + 3y = 13, xy = 6
⟹ 8x3 + 27y3 + 18(6)(13) = 2197
⟹ 8x3 + 27y3 + 1404 = 2197
⟹ 8x3 + 27y3 = 2197 – 1404
⟹ 8x3 + 27y3 = 793
Hence, the value of 8x3 + 27y3 = 793
If 3x - 2y = 11 and xy = 12, Find the value of 27x3 - 8y3
Given, 3x - 2y = 11, xy = 12
We know that (a – b)3 = a3 – b3 – 3ab(a + b)
(3x - 2y)3 = 113
⟹ 27x3 – 8y3 – (18 * 12 * 11) = 1331
⟹ 27x3 – 8y3 – 2376 = 1331
⟹ 27x3 – 8y3 = 1331 + 2376
⟹ 27x3 – 8y3 = 3707
Hence, the value of 27x3 – 8y3 = 3707
If x4 + (1/x4) = 119, Find the value of x3 − (1/x3)
Given, x4 + (1/x4) = 119 .... 1
We know that (x + y)2 = x2 + y2 + 2xy
Substitute x4 + (1/x4) = 119 in eq 1
(x2 + (1/x2))2 = x4 + (1/x4) + (2*x2* 1/x2)
= x4 + (1/x4) + 2
= 119 + 2
= 121
(x2 + (1/x2))2 = 121
x2 + (1/x2) = ±11
Now, find (x - 1/x)
We know that (x - y)2 = x2 + y2 - 2xy
(x - 1/x)2 = x2 + 1/x2 - (2*x*1/x
= x2 + 1/x2 - 2
= 11- 2
= 9
(x – 1/x) = √9
= ±3
We need to find x3 − (1/x3)
We know that, a3 - b3 = (a - b)(a2 + b2 - ab)
x3 − (1/x3) = (x - 1/x)(x2 + (1/x2) + x * 1/x
Here, x2 + (1/x2) = 11 and (x - 1/x) = 3
x3 − (1/x3) = 3(11 + 1)
= 3(12)
= 36
Hence, the value of x3 − (1/x3) = 36
Evaluate each of the following
(a) (103)3
(b) (98)3
(c) (9.9)3
(d) (10.4)3
(e) (598)3
(f) (99)3
we know that (a + b)3 = a3 + b3 + 3ab(a + b)
⟹ (103)3 can be written as (100 + 3)3
Here, a = 100 and b = 3
(103)3 = (100 + 3)3
= (100)3 + (3)3 + 3(100)(3)(100 + 3)
= 1000000 + 27 + (900*103)
= 1000000 + 27 + 92700
= 1092727
The value of (103)3 = 1092727
we know that (a - b)3 = a3 - b3 - 3ab(a - b)
⟹ (98)3 can be written as (100 - 2)3
Here, a = 100 and b = 2
(98)3 = (100 - 2)3
= (100)3 - (2)3 - 3(100)(2)(100 - 2)
= 1000000 - 8 - (600*102)
= 1000000 – 8 – 58800
= 941192
The value of (98)3 = 941192
⟹ (9.9)3 can be written as (10 – 0.1)3
Here, a = 10 and b = 0.1
(9.9)3 = (10 – 0.1)3
= (10)3 - (0.1)3 - 3(10)(0.1)(10 – 0.1)
= 1000 – 0.001 - (3*9.9)
= 1000 – 0.001 – 29.7
= 1000 – 29.701
= 970.299
The value of (9.9)3 = 970.299
⟹ (10.4)3 can be written as (10 + 0.4)3
Here, a = 10 and b = 0.4
(10.4)3 = (10 + 0.4)3
= (10)3 + (0.4)3 + 3(10)(0.4)(10 + 0.4)
= 1000 + 0.064 + (12*10.4)
= 1000 + 0.064 + 124.8
= 1000 + 124.864
= 1124.864
The value of (10.4)3 = 1124.864
⟹ (598)3 can be written as (600 - 2)3
Here, a = 600 and b = 2
(598)3 = (600 - 2)3
= (600)3 - (2)3 - 3(600)(2)(600 - 2)
= 216000000 - 8 - (3600*598)
= 216000000 - 8 - 2152800
= 216000000 - 2152808
= 213847192
The value of (598)3 = 213847192
⟹ (99)3 can be written as (100 - 1)3
Here , a = 100 and b = 1
(99)3 = (100 - 1)3
= (100)3 - (1)3 - 3(100)(1)(100 - 1)
= 1000000 - 1 - (300*99)
= 1000000 - 1 - 29700
= 1000000 - 29701
= 970299
The value of (99)3 = 970299
(a) 1113 - 893
(b) 463 + 343
(c) 1043 + 963
(d) 933 - 1073
1113 - 893
the above equation can be written as (100 + 11)3 - (100 - 11)3
we know that, (a + b)3 - (a - b)3 = 2[b3 + 3ab2]
here, a = 100 b = 11
(100 + 11)3 - (100 - 11)3 = 2[113 + 3(100)2(11)]
= 2[1331 + 330000]
= 2[331331]
= 662662
The value of 1113 - 893 = 662662
the above equation can be written as (40 + 6)3 + (40 - 6)3
we know that, (a + b)3 + (a - b)3 = 2[a3 + 3ab2]
here, a= 40 , b = 4
(40 + 6)3 + (40 - 6)3 = 2[403 + 3(6)2(40)]
= 2[64000 + 4320]
= 2[68320]
= 1366340
The value of 463 + 343 = 1366340
the above equation can be written as (100 + 4)3 + (100 - 4)3
here, a= 100 b = 4
(100 + 4)3 - (100 - 4)3 = 2[1003 + 3(4)2(100)]
= 2[1000000 + 4800]
= 2[1004800]
= 2009600
The value of 1043 + 963 = 2009600
the above equation can be written as (100 - 7)3 - (100 + 7)3
we know that, (a - b)3 - (a + b)3 = -2[b3 + 3ba2]
here, a = 93, b = 107
(100 - 7)3 - (100 + 7)3 = - 2[73 + 3(100)2(7)]
= - 2[343 + 210000]
= - 2[210343]
= - 420686
The value of 933 - 1073 = - 420686
If x + 1/x = 3, calculate x2 + 1/x2, x3 + 1/x3, x4 + 1/x4
Given, x + 1/x = 3
(x + 1/x)2 = x2 + 1/x2 + (2∗x∗1/x)
32 = x2 + 1/x2 + 2
9 - 2 = x2 + 1/x2
x2 + 1/x2 = 7
Squaring on both sides
(x2 + 1/x2)2 = 72
x4 + 1/x4 + 2* x2 * 1/x2 = 49
x4 + 1/x4 + 2 = 49
x4 + 1/x4 = 49 - 2
x4 + 1/x4 = 47
Again, cubing on both sides
(x + 1/x)3 = 33
x3 + 1/x3 + 3x*1/x(x + 1/x) = 27
x3 + 1/x3 + (3*3) = 27
x3 + 1/x3 + 9 = 27
x3 + 1/x3 = 27 - 9
x3 + 1/x3 = 18
Hence, the values are x2 + 1/x2 = 7, x4 + 1/x4 = 47, x3 + 1/x3 = 18
If x4 + 1/x4 = 194, calculate x2 + 1/x2, x3 + 1/x3, x + 1/x
x4 + 1/x4 = 194 ... 1
add and subtract (2*x2∗1/x2) on left side in above given equation
x4 + 1/x4 + (2*x2∗1/x2) - 2(2*x2∗1/x2) = 194
x4 + 1/x4 + (2*x2∗1/x2) - 2 = 194
(x2 + 1/x2)2 - 2 = 194
(x2 + 1/x2)2 = 194 + 2
(x2 + 1/x2)2 = 196
(x2 + 1/x2) = 14 ... 2
Add and subtract (2*x* 1/x) on left side in eq 2
(x2 + 1/x2) + (2*x* 1/x) - (2*x* 1/x) = 14
(x + 1/x)2 - 2 = 14
(x + 1/x)2 = 14 + 2
(x + 1/x)2 = 16
(x + 1/x) = √16
(x + 1/x) = 4 ... 3
Now, cubing eq 3 on both sides
(x + 1/x)3 = 43
We know that, (a + b)3 = a3 + b3 + 3ab(a + b)
x3 + 1/x3 + 3*x*1/x(x + 1/x) = 64
x3 + 1/x3 + (3*4) = 64
x3 + 1/x3 = 64 - 12
x3 + 1/x3 = 52
hence, the values of (x2 + 1/x2)2 = 196, (x + 1/x) = 4, x3 + 1/x3 = 52
Find the values of 27x3 + 8y3, if
(a) 3x + 2y = 14 and xy = 8
(b) 3x + 2y = 20 and xy = 14/9
(a) Given, 3x + 2y = 14 and xy = 8
cubing on both sides
(3x + 2y)3 = 143
27x3 + 8y3 + 3(3x)(2y)(3x + 2y) = 2744
27x3 + 8y3 + 18xy(3x + 2y) = 2744
27x3 + 8y3 + 18(8)(14) = 2744
27x3 + 8y3 + 2016 = 2744
27x3 + 8y3 = 2744 - 2016
27x3 + 8y3 = 728
Hence, the value of 27x3 + 8y3 = 728
(b) Given, 3x + 2y = 20 and xy = 14/9
(3x + 2y)3 = 203
27x3 + 8y3 + 3(3x)(2y)(3x + 2y) = 8000
27x3 + 8y3 + 18xy(3x + 2y) = 8000
27x3 + 8y3 + 18(14/9)(20) = 8000
27x3 + 8y3 + 560 = 8000
27x3 + 8y3 = 8000 - 560
27x3 + 8y3 = 7440
Hence, the value of 27x3 + 8y3 = 7440
Find the value of 64x3 - 125z3, if 4x - 5z = 16 and xz = 12
Given, 64x3 - 125z3
Here, 4x - 5z = 16 and xz = 12
Cubing 4x - 5z = 16 on both sides
(4x - 5z)3 = 163
We know that, (a - b)3 = a3 - b3 - 3ab(a - b)
(4x)3 - (5z)3 - 3(4x)(5z)(4x - 5z) = 163
64x3 - 125z3 - 60(xz)(16) = 4096
64x3 - 125z3 - 60(12)(16) = 4096
64x3 - 125z3 - 11520 = 4096
64x3 - 125z3 = 4096 + 11520
64x3 - 125z3 = 15616
The value of 64x3 - 125z3 = 15616
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Chapter 4: Algebraic Identities Exercise –...