The problem asks us to simplify the given expressions by removing the brackets. The expressions are: (a) $(a4^3)^2$ (b) $(x^5)^3$ (c) $(a^2b^3)^4$ (d) $a^2(a^3 + a^5)$ (e) $2x^4(3x - 5x^7)$ (f) $m^2(1 - m) - 2m(m + 2m^2)$ (g) $ab(a^2 + ab - b^2)$

1. Problem Description

The problem asks us to simplify the given expressions by removing the brackets. The expressions are:

(a) $(a4^3)^2$

(b) $(x^5)^3$

(c) $(a^2b^3)^4$

(d) $a^2(a^3 + a^5)$

(e) $2x^4(3x - 5x^7)$

(f) $m^2(1 - m) - 2m(m + 2m^2)$

(g) $ab(a^2 + ab - b^2)$

2. Solution Steps

(a) $(a4^3)^2 = (a \cdot 64)^2 = (64a)^2 = 64^2 \cdot a^2 = 4096a^2$

(b) $(x^5)^3$

$(x^a)^b = x^{a \cdot b}$

$(x^5)^3 = x^{5 \cdot 3} = x^{15}$

(c) $(a^2b^3)^4$

$(ab)^c = a^c b^c$

$(a^2b^3)^4 = (a^2)^4 (b^3)^4 = a^{2 \cdot 4} b^{3 \cdot 4} = a^8b^{12}$

(d) $a^2(a^3 + a^5)$

$a(b+c) = ab + ac$

$a^2(a^3 + a^5) = a^2 \cdot a^3 + a^2 \cdot a^5 = a^{2+3} + a^{2+5} = a^5 + a^7$

(e) $2x^4(3x - 5x^7)$

$2x^4(3x - 5x^7) = 2x^4 \cdot 3x - 2x^4 \cdot 5x^7 = 6x^{4+1} - 10x^{4+7} = 6x^5 - 10x^{11}$

(f) $m^2(1 - m) - 2m(m + 2m^2)$

$m^2(1 - m) - 2m(m + 2m^2) = m^2 \cdot 1 - m^2 \cdot m - (2m \cdot m + 2m \cdot 2m^2) = m^2 - m^3 - (2m^2 + 4m^3) = m^2 - m^3 - 2m^2 - 4m^3 = (m^2 - 2m^2) + (-m^3 - 4m^3) = -m^2 - 5m^3$

(g) $ab(a^2 + ab - b^2)$

$ab(a^2 + ab - b^2) = ab \cdot a^2 + ab \cdot ab - ab \cdot b^2 = a^3b + a^2b^2 - ab^3$

3. Final Answer

(a) $4096a^2$

(b) $x^{15}$

(c) $a^8b^{12}$

(d) $a^5 + a^7$

(e) $6x^5 - 10x^{11}$

(f) $-m^2 - 5m^3$

(g) $a^3b + a^2b^2 - ab^3$