The problem asks us to simplify the following expressions: (b) $(x^5)^3$ (c) $(a^2b^3)^4$ (d) $a^2(a^3+a^5)$ (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 following expressions:

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

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

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

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

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

2. Solution Steps

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

Using the power of a power rule $(x^a)^b = x^{a*b}$, we get

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

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

Using the power of a product rule $(xy)^a = x^a y^a$, we have

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

Using the power of a power rule $(x^a)^b = x^{a*b}$, we have

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

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

Using the distributive property $a(b+c) = ab+ac$, we have

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

Using the product of powers rule $x^a x^b = x^{a+b}$, we have

$a^2a^3+a^2a^5 = a^{2+3}+a^{2+5} = a^5+a^7$

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

Using the distributive property $a(b+c) = ab+ac$, we have

$m^2(1-m) = m^2 - m^3$

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

Thus, we have

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

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

Using the distributive property $a(b+c+d) = ab+ac+ad$, we have

$ab(a^2+ab-b^2) = aba^2+ab(ab)-ab(b^2)$

Using the product of powers rule $x^a x^b = x^{a+b}$, we have

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

3. Final Answer

(b) $x^{15}$

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

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

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

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