The problem is to simplify algebraic 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 is to simplify algebraic 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) $(a4^3)^2 = (a64)^2 = a^2 64^2 = 4096a^2$

(b) $(x^5)^3 = x^{5*3} = x^{15}$

The power of a power property states that $(a^m)^n = a^{mn}$.

(c) $(a^2b^3)^4 = a^{2*4}b^{3*4} = a^8b^{12}$

The power of a product property states that $(ab)^n = a^nb^n$.

(d) $a^2(a^3+a^5) = a^2*a^3 + a^2*a^5 = a^{2+3} + a^{2+5} = a^5+a^7$

The product of powers property states that $a^m * a^n = a^{m+n}$.

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

(f) $m^2(1-m)-2m(m+2m^2) = m^2*1 - m^2*m - 2m*m - 2m*2m^2 = m^2 - m^3 - 2m^2 - 4m^3 = m^2-2m^2-m^3-4m^3 = -m^2 - 5m^3$

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

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