Simplify the expressions by removing the brackets. The given expressions are: (a) $(a4^3)^2$ (b) $(x^5)^3$ (c) $(a^2b^3)^4$ (e) $2x^4(3x-5x^7)$ (f) $m^2(1-m)-2m(m+2m^2)$

1. Problem Description

Simplify the expressions by removing the brackets. The given expressions are:

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

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

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

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

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

2. Solution Steps

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

Here, we assume the expression intended is $(a(4^3))^2$.

$4^3 = 4 \times 4 \times 4 = 64$

So, the expression becomes $(64a)^2 = (64)^2 \times a^2 = 4096a^2$

Assuming the expression intended is $(a \cdot 4^3)^2$, the result is $4096a^2$.

If we assume the intended expression is $(a^4)^3$, then $(a^4)^3 = a^{4 \times 3} = a^{12}$. Then the expression becomes $(a^{12})^2 = a^{12 \times 2} = a^{24}$.

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

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

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

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

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

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

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

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

3. Final Answer

(a) $4096a^2$ or $a^{24}$ (depending on interpretation)

(b) $x^{15}$

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

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

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