Exercise 4.2: Algebraic Formulae and Problems Related to Cubes
1. Find the cube of the following expressions with the help of formula:
Required Formula:
- $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
- $(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3$
- $(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)$
(a) $3x + y$
Solution: $(3x + y)^3$
$= (3x)^3 + 3 \cdot (3x)^2 \cdot y + 3 \cdot 3x \cdot y^2 + y^3$
$= 27x^3 + 3 \cdot 9x^2 \cdot y + 9xy^2 + y^3$
$= 27x^3 + 27x^2y + 9xy^2 + y^3$ (Answer)
(b) $x^2 + y$
Solution: $(x^2 + y)^3$
$= (x^2)^3 + 3 \cdot (x^2)^2 \cdot y + 3 \cdot x^2 \cdot y^2 + y^3$
$= x^6 + 3x^4y + 3x^2y^2 + y^3$ (Answer)
(c) $5p + 2q$
Solution: $(5p + 2q)^3$
$= (5p)^3 + 3 \cdot (5p)^2 \cdot 2q + 3 \cdot 5p \cdot (2q)^2 + (2q)^3$
$= 125p^3 + 3 \cdot 25p^2 \cdot 2q + 3 \cdot 5p \cdot 4q^2 + 8q^3$
$= 125p^3 + 150p^2q + 60pq^2 + 8q^3$ (Answer)
(d) $a^2b + c^2d$
Solution: $(a^2b + c^2d)^3$
$= (a^2b)^3 + 3 \cdot (a^2b)^2 \cdot c^2d + 3 \cdot a^2b \cdot (c^2d)^2 + (c^2d)^3$
$= a^6b^3 + 3a^4b^2c^2d + 3a^2bc^4d^2 + c^6d^3$ (Answer)
(e) $6p – 7$
Solution: $(6p – 7)^3$
$= (6p)^3 – 3 \cdot (6p)^2 \cdot 7 + 3 \cdot 6p \cdot (7)^2 – (7)^3$
$= 216p^3 – 3 \cdot 36p^2 \cdot 7 + 3 \cdot 6p \cdot 49 – 343$
$= 216p^3 – 756p^2 + 882p – 343$ (Answer)
(f) $ax – by$
Solution: $(ax – by)^3$
$= (ax)^3 – 3 \cdot (ax)^2 \cdot by + 3 \cdot ax \cdot (by)^2 – (by)^3$
$= a^3x^3 – 3a^2b x^2 y + 3ab^2 x y^2 – b^3y^3$ (Answer)
(g) $2p^2 – 3r^2$
Solution: $(2p^2 – 3r^2)^3$
$= (2p^2)^3 – 3 \cdot (2p^2)^2 \cdot 3r^2 + 3 \cdot 2p^2 \cdot (3r^2)^2 – (3r^2)^3$
$= 8p^6 – 3 \cdot 4p^4 \cdot 3r^2 + 3 \cdot 2p^2 \cdot 9r^4 – 27r^6$
$= 8p^6 – 36p^4r^2 + 54p^2r^4 – 27r^6$ (Answer)
(h) $x^3 + 2$
Solution: $(x^3 + 2)^3$
$= (x^3)^3 + 3 \cdot (x^3)^2 \cdot 2 + 3 \cdot x^3 \cdot (2)^2 + (2)^3$
$= x^9 + 6x^6 + 12x^3 + 8$ (Answer)
(i) $2m + 3n – 5p$
Solution: $(2m + 3n – 5p)^3$
$= \{(2m + 3n) – 5p\}^3$
$= (2m + 3n)^3 – 3 \cdot (2m + 3n)^2 \cdot 5p + 3 \cdot (2m + 3n) \cdot (5p)^2 – (5p)^3$
$= \{(2m)^3 + 3(2m)^2(3n) + 3(2m)(3n)^2 + (3n)^3\} – 15p(4m^2 + 12mn + 9n^2) + 3(2m + 3n)(25p^2) – 125p^3$
$= (8m^3 + 36m^2n + 54mn^2 + 27n^3) – (60m^2p + 180mnp + 135n^2p) + 75p^2(2m + 3n) – 125p^3$
$= 8m^3 + 27n^3 – 125p^3 + 36m^2n + 54mn^2 – 60m^2p – 135n^2p + 150mp^2 + 225np^2 – 180mnp$ (Answer)
(j) $x^2 – y^2 + z^2$
Solution: $(x^2 – y^2 + z^2)^3$
$= \{(x^2 – y^2) + z^2\}^3$
$= (x^2 – y^2)^3 + 3(x^2 – y^2)^2(z^2) + 3(x^2 – y^2)(z^2)^2 + (z^2)^3$
$= \{(x^2)^3 – 3(x^2)^2(y^2) + 3(x^2)(y^2)^2 – (y^2)^3\} + 3z^2(x^4 – 2x^2y^2 + y^4) + 3z^4(x^2 – y^2) + z^6$
$= x^6 – 3x^4y^2 + 3x^2y^4 – y^6 + 3x^4z^2 – 6x^2y^2z^2 + 3y^4z^2 + 3x^2z^4 – 3y^2z^4 + z^6$
$= x^6 – y^6 + z^6 – 3x^4y^2 + 3x^2y^4 + 3x^4z^2 + 3y^4z^2 + 3x^2z^4 – 3y^2z^4 – 6x^2y^2z^2$ (Answer)
(k) $a^2b^2 – c^2d^2$
Solution: $(a^2b^2 – c^2d^2)^3$
$= (a^2b^2)^3 – 3 \cdot (a^2b^2)^2 \cdot c^2d^2 + 3 \cdot a^2b^2 \cdot (c^2d^2)^2 – (c^2d^2)^3$
$= a^6b^6 – 3a^4b^4c^2d^2 + 3a^2b^2c^4d^4 – c^6d^6$ (Answer)
(l) $a^2b – b^3c$
Solution: $(a^2b – b^3c)^3$
$= (a^2b)^3 – 3 \cdot (a^2b)^2 \cdot b^3c + 3 \cdot a^2b \cdot (b^3c)^2 – (b^3c)^3$
$= a^6b^3 – 3a^4b^2 \cdot b^3c + 3a^2b \cdot b^6c^2 – b^9c^3$
$= a^6b^3 – 3a^4b^5c + 3a^2b^7c^2 – b^9c^3$ (Answer)
(m) $x^3 – 2y^3$
Solution: $(x^3 – 2y^3)^3$
$= (x^3)^3 – 3 \cdot (x^3)^2 \cdot 2y^3 + 3 \cdot x^3 \cdot (2y^3)^2 – (2y^3)^3$
$= x^9 – 3 \cdot x^6 \cdot 2y^3 + 3 \cdot x^3 \cdot 4y^6 – 8y^9$
$= x^9 – 6x^6y^3 + 12x^3y^6 – 8y^9$ (Answer)
(n) $11a – 12b$
Solution: $(11a – 12b)^3$
$= (11a)^3 – 3 \cdot (11a)^2 \cdot 12b + 3 \cdot 11a \cdot (12b)^2 – (12b)^3$
$= 1331a^3 – 3 \cdot 121a^2 \cdot 12b + 3 \cdot 11a \cdot 144b^2 – 1728b^3$
$= 1331a^3 – 4356a^2b + 4752ab^2 – 1728b^3$ (Answer)
(o) $x^3 + y^3$
Solution: $(x^3 + y^3)^3$
$= (x^3)^3 + 3 \cdot (x^3)^2 \cdot y^3 + 3 \cdot x^3 \cdot (y^3)^2 + (y^3)^3$
$= x^9 + 3x^6y^3 + 3x^3y^6 + y^9$ (Answer)
2. Simplify:
(a) $(3x + y)^3 + 3(3x + y)^2(3x – y) + 3(3x + y)(3x – y)^2 + (3x – y)^3$
Solution: Let, $3x + y = a$ and $3x – y = b$
Given expression $= (3x + y)^3 + 3(3x + y)^2(3x – y) + 3(3x + y)(3x – y)^2 + (3x – y)^3$
$= a^3 + 3a^2b + 3ab^2 + b^3$
$= (a + b)^3$
$= \{(3x + y) + (3x – y)\}^3$ [Substituting the values of $a$ and $b$]
$= (3x + y + 3x – y)^3$
$= (6x)^3$
$= 216x^3$ (Answer)
(b) $(2p + 5q)^3 + 3(2p + 5q)^2(5q – 2p) + 3(2p + 5q)(5q – 2p)^2 + (5q – 2p)^3$
Solution: Let, $2p + 5q = a$ and $5q – 2p = b$
Given expression $= (2p + 5q)^3 + 3(2p + 5q)^2(5q – 2p) + 3(2p + 5q)(5q – 2p)^2 + (5q – 2p)^3$
$= a^3 + 3a^2b + 3ab^2 + b^3$
$= (a + b)^3$
$= \{(2p + 5q) + (5q – 2p)\}^3$ [Substituting the values of $a$ and $b$]
$= (2p + 5q + 5q – 2p)^3$
$= (10q)^3$
$= 1000q^3$ (Answer)
(c) $(x + 2y)^3 – 3(x + 2y)^2(x – 2y) + 3(x + 2y)(x – 2y)^2 – (x – 2y)^3$
Solution: Let, $x + 2y = a$ and $x – 2y = b$
Given expression $= (x + 2y)^3 – 3(x + 2y)^2(x – 2y) + 3(x + 2y)(x – 2y)^2 – (x – 2y)^3$
$= a^3 – 3a^2b + 3ab^2 – b^3$
$= (a – b)^3$
$= \{(x + 2y) – (x – 2y)\}^3$ [Substituting the values of $a$ and $b$]
$= (x + 2y – x + 2y)^3$
$= (4y)^3$
$= 64y^3$ (Answer)
(d) $(6m + 2)^3 – 3(6m + 2)^2(6m – 4) + 3(6m + 2)(6m – 4)^2 – (6m – 4)^3$
Solution: Let, $6m + 2 = a$ and $6m – 4 = b$
Given expression $= (6m + 2)^3 – 3(6m + 2)^2(6m – 4) + 3(6m + 2)(6m – 4)^2 – (6m – 4)^3$
$= a^3 – 3a^2b + 3ab^2 – b^3$
$= (a – b)^3$
$= \{(6m + 2) – (6m – 4)\}^3$ [Substituting the values of $a$ and $b$]
$= (6m + 2 – 6m + 4)^3$
$= (6)^3$
$= 216$ (Answer)
(e) $(x – y)^3 + (x + y)^3 + 6x(x^2 – y^2)$
Solution: Given expression $= (x – y)^3 + (x + y)^3 + 6x(x^2 – y^2)$
$= (x – y)^3 + (x + y)^3 + 3 \cdot 2x \cdot (x + y)(x – y)$
$= (x + y)^3 + (x – y)^3 + 3(x + y)(x – y)\{(x + y) + (x – y)\}$
Let, $x + y = a$ and $x – y = b$
$\therefore a + b = (x + y) + (x – y) = 2x$
Given expression $= a^3 + b^3 + 3ab(a + b)$
$= (a + b)^3$
$= (2x)^3$ [Substituting the value of $a + b$]
$= 8x^3$ (Answer)
3. If $a + b = 8$ and $ab = 15$, what is the value of $a^3 + b^3$?
Solution: Given, $a + b = 8$ and $ab = 15$
Given expression $= a^3 + b^3$
$= (a + b)^3 – 3ab(a + b)$
$= (8)^3 – 3 \cdot 15 \cdot 8$
$= 512 – 360$
$= 152$ (Answer)
4. If $x + y = 2$, show that $x^3 + y^3 + 6xy = 8$.
Solution: Given, $x + y = 2$
L.H.S. $= x^3 + y^3 + 6xy$
$= (x + y)^3 – 3xy(x + y) + 6xy$
$= (2)^3 – 3xy(2) + 6xy$
$= 8 – 6xy + 6xy$
$= 8$
$= \text{R.H.S.}$
$\therefore x^3 + y^3 + 6xy = 8$ (Shown)
5. If $2x + 3y = 13$ and $xy = 6$, find the value of $8x^3 + 27y^3$.
Solution: Given, $2x + 3y = 13$ and $xy = 6$
Given expression $= 8x^3 + 27y^3$
$= (2x)^3 + (3y)^3$
$= (2x + 3y)^3 – 3 \cdot 2x \cdot 3y(2x + 3y)$
$= (2x + 3y)^3 – 18xy(2x + 3y)$
$= (13)^3 – 18 \cdot 6 \cdot 13$
$= 2197 – 1404$
$= 793$ (Answer)
6. If $p – q = 5, pq = 3$, find the value of $p^3 – q^3$.
Solution: Given, $p – q = 5$ and $pq = 3$
Given expression $= p^3 – q^3$
$= (p – q)^3 + 3pq(p – q)$
$= (5)^3 + 3 \cdot 3 \cdot 5$
$= 125 + 45$
$= 170$ (Answer)
7. If $x – 2y = 3$, find the value of $x^3 – 8y^3 – 18xy$.
Solution: Given, $x – 2y = 3$
Given expression $= x^3 – 8y^3 – 18xy$
$= (x)^3 – (2y)^3 – 18xy$
$= (x – 2y)^3 + 3 \cdot x \cdot 2y(x – 2y) – 18xy$
$= (x – 2y)^3 + 6xy(x – 2y) – 18xy$
$= (3)^3 + 6xy(3) – 18xy$
$= 27 + 18xy – 18xy$
$= 27$ (Answer)
8. If $4x – 3 = 5$, prove that $64x^3 – 27 – 180x = 125$.
Solution: Given, $4x – 3 = 5$
L.H.S. $= 64x^3 – 27 – 180x$
$= (4x)^3 – (3)^3 – 180x$
$= (4x – 3)^3 + 3 \cdot 4x \cdot 3(4x – 3) – 180x$
$= (4x – 3)^3 + 36x(4x – 3) – 180x$
$= (5)^3 + 36x(5) – 180x$
$= 125 + 180x – 180x$
$= 125$
$= \text{R.H.S.}$
$\therefore 64x^3 – 27 – 180x = 125$ (Proved)
9. If $a = -3$ and $b = 2$, find the value of $8a^3 + 36a^2b + 54ab^2 + 27b^3$.
Solution: Given, $a = -3$ and $b = 2$
Given expression $= 8a^3 + 36a^2b + 54ab^2 + 27b^3$
$= (2a)^3 + 3 \cdot (2a)^2 \cdot 3b + 3 \cdot 2a \cdot (3b)^2 + (3b)^3$
$= (2a + 3b)^3$
$= \{2(-3) + 3(2)\}^3$ [Substituting the values of $a$ and $b$]
$= (-6 + 6)^3$
$= (0)^3$
$= 0$ (Answer)
10. If $a = 7$, find the value of $a^3 + 6a^2 + 12a + 1$.
Solution: Given, $a = 7$
Given expression $= a^3 + 6a^2 + 12a + 1$
$= a^3 + 3 \cdot a^2 \cdot 2 + 3 \cdot a \cdot (2)^2 + (2)^3 – 7$
$= (a + 2)^3 – 7$
$= (7 + 2)^3 – 7$ [Substituting the value of $a$]
$= (9)^3 – 7$
$= 729 – 7$
$= 722$ (Answer)
11. If $x = 5$, what is the value of $x^3 – 12x^2 + 48x – 64$?
Solution: Given, $x = 5$
Given expression $= x^3 – 12x^2 + 48x – 64$
$= x^3 – 3 \cdot x^2 \cdot 4 + 3 \cdot x \cdot (4)^2 – (4)^3$
$= (x – 4)^3$
$= (5 – 4)^3$ [Substituting the value of $x$]
$= (1)^3$
$= 1$ (Answer)
12. If $a^2 + b^2 = c^2$, prove that $a^6 + b^6 + 3a^2b^2c^2 = c^6$.
Solution: Given, $a^2 + b^2 = c^2$
L.H.S. $= a^6 + b^6 + 3a^2b^2c^2$
$= (a^2)^3 + (b^2)^3 + 3a^2b^2c^2$
$= (a^2 + b^2)^3 – 3a^2b^2(a^2 + b^2) + 3a^2b^2c^2$
$= (c^2)^3 – 3a^2b^2(c^2) + 3a^2b^2c^2$
$= c^6 – 3a^2b^2c^2 + 3a^2b^2c^2$
$= c^6$
$= \text{R.H.S.}$
$\therefore a^6 + b^6 + 3a^2b^2c^2 = c^6$ (Proved)
13. If $x + \frac{1}{x} = 4$, prove that $x^3 + \frac{1}{x^3} = 52$.
Solution: Given, $x + \frac{1}{x} = 4$
L.H.S. $= x^3 + \frac{1}{x^3}$
$= \left(x + \frac{1}{x}\right)^3 – 3 \cdot x \cdot \frac{1}{x}\left(x + \frac{1}{x}\right)$
$= (4)^3 – 3(4)$
$= 64 – 12$
$= 52$
$= \text{R.H.S.}$
$\therefore x^3 + \frac{1}{x^3} = 52$ (Proved)
14. If $a – \frac{1}{a} = 5$, what is the value of $a^3 – \frac{1}{a^3}$?
Solution: Given, $a – \frac{1}{a} = 5$
Given expression $= a^3 – \frac{1}{a^3}$
$= \left(a – \frac{1}{a}\right)^3 + 3 \cdot a \cdot \frac{1}{a}\left(a – \frac{1}{a}\right)$
$= (5)^3 + 3(5)$
$= 125 + 15$
$= 140$ (Answer)
15. Find the product with the help of formula:
Required Formulae:
- $(a + b)(a^2 – ab + b^2) = a^3 + b^3$
- $(a – b)(a^2 + ab + b^2) = a^3 – b^3$
(a) $(a^2 + b^2)(a^4 – a^2b^2 + b^4)$
Solution: $(a^2 + b^2)(a^4 – a^2b^2 + b^4)$
$= (a^2 + b^2)\{(a^2)^2 – a^2 \cdot b^2 + (b^2)^2\}$
$= (a^2)^3 + (b^2)^3$
$= a^6 + b^6$ (Answer)
(b) $(ax – by)(a^2x^2 + abxy + b^2y^2)$
Solution: $(ax – by)(a^2x^2 + abxy + b^2y^2)$
$= (ax – by)\{(ax)^2 + ax \cdot by + (by)^2\}$
$= (ax)^3 – (by)^3$
$= a^3x^3 – b^3y^3$ (Answer)
(c) $(2ab^2 – 1)(4a^2b^4 + 2ab^2 + 1)$
Solution: $(2ab^2 – 1)(4a^2b^4 + 2ab^2 + 1)$
$= (2ab^2 – 1)\{(2ab^2)^2 + 2ab^2 \cdot 1 + (1)^2\}$
$= (2ab^2)^3 – (1)^3$
$= 8a^3b^6 – 1$ (Answer)
(d) $(x^2 + a)(x^4 – ax^2 + a^2)$
Solution: $(x^2 + a)(x^4 – ax^2 + a^2)$
$= (x^2 + a)\{(x^2)^2 – x^2 \cdot a + a^2\}$
$= (x^2)^3 + a^3$
$= x^6 + a^3$ (Answer)
(e) $(7a + 4b)(49a^2 – 28ab + 16b^2)$
Solution: $(7a + 4b)(49a^2 – 28ab + 16b^2)$
$= (7a + 4b)\{(7a)^2 – 7a \cdot 4b + (4b)^2\}$
$= (7a)^3 + (4b)^3$
$= 343a^3 + 64b^3$ (Answer)
(f) $(2a – 1)(4a^2 + 2a + 1)(8a^3 + 1)$
Solution: $(2a – 1)(4a^2 + 2a + 1)(8a^3 + 1)$
$= (2a – 1)\{(2a)^2 + 2a \cdot 1 + (1)^2\}(8a^3 + 1)$
$= \{(2a)^3 – (1)^3\}(8a^3 + 1)$
$= (8a^3 – 1)(8a^3 + 1)$
$= (8a^3)^2 – (1)^2$
$= 64a^6 – 1$ (Answer)
(g) $(x + a)(x^2 – ax + a^2)(x – a)(x^2 + ax + a^2)$
Solution: $(x + a)(x^2 – ax + a^2)(x – a)(x^2 + ax + a^2)$
$= \{(x + a)(x^2 – ax + a^2)\} \times \{(x – a)(x^2 + ax + a^2)\}$
$= (x^3 + a^3)(x^3 – a^3)$
$= (x^3)^2 – (a^3)^2$
$= x^6 – a^6$ (Answer)
(h) $(5a + 3b)(25a^2 – 15ab + 9b^2)(125a^3 – 27b^3)$
Solution: $(5a + 3b)(25a^2 – 15ab + 9b^2)(125a^3 – 27b^3)$
$= (5a + 3b)\{(5a)^2 – 5a \cdot 3b + (3b)^2\}(125a^3 – 27b^3)$
$= \{(5a)^3 + (3b)^3\}(125a^3 – 27b^3)$
$= (125a^3 + 27b^3)(125a^3 – 27b^3)$
$= (125a^3)^2 – (27b^3)^2$
$= 15625a^6 – 729b^6$ (Answer)