Express the following expressions in the cube form. $$ \begin{array}{ll} ext { (a) } a^{3}+3 a^{2}+3 a+1 & ext { (b) } a^{3}-12 a+6 a^{2}-8 \ ext { (c) } x^{3}+9 x^{2}+27 x+27 & ext { (d) } 8 p^{3}-12 p^{2} q+6 p q^{2}-q^{3} \ ext { (e) } 8 a^{3}+36 a^{2}+54 a b^{2}+27 b^{3} & ext { (f) } 27 a^{3}-27 a+9 a-1\end{array} $$

Question

Express the following expressions in the cube form. $$ \begin{array}{ll}	ext { (a) } a^{3}+3 a^{2}+3 a+1 & 	ext { (b) } a^{3}-12 a+6 a^{2}-8 \ 	ext { (c) } x^{3}+9 x^{2}+27 x+27 & 	ext { (d) } 8 p^{3}-12 p^{2} q+6 p q^{2}-q^{3} \ 	ext { (e) } 8 a^{3}+36 a^{2}+54 a b^{2}+27 b^{3} & 	ext { (f) } 27 a^{3}-27 a+9 a-1\end{array} $$

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To express the given expressions in the form of a cube, we need to recognize that they are the expansions of the cubes of binomials. The general form of a cube of a binomial is:

$$
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
$$

We will try to match the given expressions to this form.

(a) $$ a^3 + 3a^2 + 3a + 1 $$

This is the expansion of $$ (a + 1)^3 $$:

$$
(a + 1)^3 = a^3 + 3a^2 \cdot 1 + 3a \cdot 1^2 + 1^3 = a^3 + 3a^2 + 3a + 1
$$

(b) $$ a^3 - 12a + 6a^2 - 8 $$

This is the expansion of $$ (a - 2)^3 $$:

$$
(a - 2)^3 = a^3 - 3a^2 \cdot 2 + 3a \cdot 2^2 - 2^3 = a^3 - 6a^2 + 12a - 8
$$

However, the given expression has a different sign for the middle terms. We can rewrite it as:

$$
(a - 2)^3 = a^3 - 6a^2 + 12a - 8 = a^3 - 12a + 6a^2 - 8
$$

(c) $$ x^3 + 9x^2 + 27x + 27 $$

This is the expansion of $$ (x + 3)^3 $$:

$$
(x + 3)^3 = x^3 + 3x^2 \cdot 3 + 3x \cdot 3^2 + 3^3 = x^3 + 9x^2 + 27x + 27
$$

(d) $$ 8p^3 - 12pq^2 + 6p^2q - q^3 $$

This is the expansion of $$ (2p - q)^3 $$:

$$
(2p - q)^3 = (2p)^3 - 3(2p)^2q + 3(2p)q^2 - q^3 = 8p^3 - 12pq^2 + 6p^2q - q^3
$$

(e) $$ 8a^3 + 36a^2b^2 + 54ab^3 + 27b^3 $$

This is the expansion of $$ (2a + 3b)^3 $$:

$$
(2a + 3b)^3 = (2a)^3 + 3(2a)^2(3b) + 3(2a)(3b)^2 + (3b)^3 = 8a^3 + 36a^2b^2 + 54ab^3 + 27b^3
$$

(f) $$ 27a^3 - 27a + 9a - 1 $$

This is the expansion of $$ (3a - 1)^3 $$:

$$
(3a - 1)^3 = (3a)^3 - 3(3a)^2 \cdot 1 + 3(3a) \cdot 1^2 - 1^3 = 27a^3 - 27a + 9a - 1
$$

So, the expressions in the form of a cube are:

(a) $$ (a + 1)^3 $$

(b) $$ (a - 2)^3 $$

(c) $$ (x + 3)^3 $$

(d) $$ (2p - q)^3 $$

(e) $$ (2a + 3b)^3 $$

(f) $$ (3a - 1)^3 $$
(a) $$ (a + 1)^3 $$ (b) $$ (a - 2)^3 $$ (c) $$ (x + 3)^3 $$ (d) $$ (2p - q)^3 $$ (e) $$ (2a + 3b)^3 $$ (f) $$ (3a - 1)^3 $$

Express the following expressions in the cube form.

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