Exercise 1c

$(3m^4)^2$

The $^2$ affects both $3$ and $m^4$ since they are both enclosed in the bracket.

$(3m^4)^2=(3 \times m^4)^2$

$=3^2 \times m^{4 \times 2}$

$=9\times m^8$

$=9m^8$

Answer: $9m^8$

$(2n^5)^3$

The $^3$ affects both $2$ and $n^5$ since they are both enclosed in the bracket.

$(2n^5)^3=(2 \times n^5)^3$

$=2^3 \times n^{5 \times 3}$

$=8\times n^{15}$

$=8n^{15}$

Answer: $8n^{15}$

$(4v^3)^2$

Recall: $(a^m)^n=a^{m \times n}$

Also note that the $^2$ affects both $4$ and $v^3$ since they are both enclosed in the bracket.

$(4v^3)^2$

$=(4 \times v^3)^2$

$=4^2 \times v^{3 \times 2}$

$=16 \times v^6$

$=16v^6$

Answer: $16v^6$

$4(v^3)^2$

Since $4$ is not in the bracket, the $^2$ only affects $v^3$.

Recall: $(a^m)^n=a^{m \times n}$

$=4 \times v^{3 \times 2}$

$=4 \times v^6$

$=4v^6$

Answer: $4v^6$

$-2(a^2)^3$

Recall: $(a^m)^n=a^{m \times n}$

$=-2 \times  a^{2 \times 3}$

$=-2 \times a^6$

$=-2a^6$

Answer: $-2a^6$

$(-2b^2)^3$

$=(-2 \times b^2)^3$

$=(-2)^3 \times b^{2 \times 3}$

$=-8 \times b^6$

$=-8b^6$

Answer: $-8b^6$

$(-c^3)^2$

Recall: $(a^m)^n$

$=(-1 \times c^3)^2$

$=(-1)^2 \times c^{3 \times 2}$

$=1 \times c^6$

$=c^6$

Answer: $c^6$

$(-e^4)^3$

Recall: $(a^m)^n = a^{m \times n}$

$=(-1 \times e^4)^3$

$=(-1)^3 \times (e^4)^3$

$=-1 \times e^{4 \times 3}$

$=-1 \times e^{12}$

$=-e^{12}$

Answer: $-e^{12}$

$(-u^2)^5$

Recall: $(a^m)^n$

$=(-u^2)^5$

$=(-1 \times u^2)^5$

$=(-1 \times u^2)^5$

$=(-1)^5 \times u^{2 \times 5}$

$=-1 \times u^{10}$

$=-u^{10}$

Answer: $-u^{10}$

Simplify $-(c^5)^4$

Recall: $(a^m)^n=a^{m \times n}$

$=-1 \times c^{5 \times 4}$

$=-1 \times c^{20}$

$=-c^{20}$

Answer: $-c^{20}$

$(-d^5)^4$

$=(-1 \times d^5)^4$

Recall: $(a^m)^n = a^{m \times n}$

$=(-1)^4 \times d^{5 \times 4}$

$=1 \times d^{20}$

$=d^{20}$

Answer $=d^{20}$

$(-f^4)^5$

Recall $(a^m)^n=a^{m \times n}$

$=(-1 \times f^4)^5$

$=(-1)^5 \times f^{4 \times 5}$

$=-1 \times f^{20}$

$=-f^{20}$

$(mn^2)^4$

Recall: $(a^m)^n = a^{m \times n}$

$=(m \times n^2)^4$

$=m^4 \times n^{2 \times 4}$

$=m^4 \times n^8$

$=m^4n^8$

Answer $=m^4n^8$ 

$(a^2b)^3$

$=(a^2 \times b)^3$

Recall: $(a^m)^n = a^{m \times n}$

$=(a^2)^3 \times b^3$

$=a^{2 \times 3} \times b^3$

$=a^6 \times b^3$

$=a^6b^3$

Answer: $a^6b^3$

$(x^2y^3)^4$

$=(x^2 \times y^3)^4$

$=(x^2)^ 4 \times (y^3)^4$

Recall: $(a^m)^n=a^{m \times n}$

$=x^{2 \times 4} \times y^{3 \times 4}$

$=x^8 \times y^{12}$

$=x^8y^{12}$

Answer: $x^8y^{12}$

$(-u^3v^3)^4$

$=(-1 \times u^3 \times v^2)^4$

Recall: $(a^m)^n=a^{m \times n}$

$=(-1)^4 \times u^{3 \times 4} \times v^{2 \times 4}$

$=1 \times u^{3 \times 4} \times v^{2 \times 4}$

$=1 \times u^{12} \times v^{8}$

$=u^{12}v^{8}$

Answer: $u^{12}v^{8}$

$(5mn^3)^3$

$=(5 \times m \times n^3)^3$

Recall: $(a^m)^n=a^{m \times n}$

$=(5 \times m \times n^3)^3$

$=5^3 \times m^3 \times n^{3 \times 3}$

$=125 \times m^3 \times n^9$

$=125m^3n^9$

Answer: $125m^3n^9$

$(-4u^2v)^3$

$=(-4 \times u^2 \times v)^3$

Recall: $(a^m)^n=a^{m\times n}$

$=(-4)^3 \times (u^2)^3 \times v^3$

$=-64 \times u^6 \times v^3$

$=-64u^6v^3$

Answer: $-64u^6v^3$

$(-a^2m)^4$

$=(-1 \times a^2 \times m)^4$

Recall: $(a^m)^n=a^{m \times n}$

$=(-1)^4 \times a^{2 \times 4} \times m^4$

$=1 \times a^8 \times m^4$

$=a^8m^4$

Answer: $a^8m^4$

$-3(de^3)^4$

$=-3 (d \times e^3)^4$

Recall: $(a^m)^n$

$=-3 \times d^4 \times e^{3 \times 4}$

$=-3 \times d^4 \times e^{12}$

$=-3d^4e^{12}$

Answer: $-3d^4e^{12}$

$\frac{(-x^3)^2}{-x^4}$

$=\frac{(-1 \times x^3)^3 }{-x^4}$

Recall: $(a^m)^n=a^{m \times n}$

$=\frac{(-1^3 \times x^{3 \times 3})}{-x^4}$

$=\frac{-1\times x^9}{-x^4}$

$=\frac{-x^9}{-x^4}$

$=\frac{x^9}{x^4}$

$=x^5$

Answer: $x^5$

$\frac{a^6}{(-a)^4}$

$=\frac{a^6}{a^4}$

Recall: $\frac{a^m}{a^n}=a^m \div a^n=a^{m-n}$

$=a^6 \div a^4 =a^{6-4}$

$=a^2$

Answer: $a^2$

$\frac{(-c)^2 \times c^4}{(-c)^5}$

$=\frac{(-1 \times c)^2 \times c^4}{(-1 \times c)^5}$

$=\frac{(-1)^2 \times c^2 \times c^4}{(-1)^5 \times c^5}$

Recall: $a^m \times a^n = a^{m+n}$

$=\frac{1 \times c^{2 + 4 }}{-1 \times c^5}$

$=\frac{c^6}{-c^5}$

$=-1 \times (c^{6-5}) $

$=-1 \times c$

$=-c$

Answer: $-c$

$\frac{-(d^2)^3}{d^4 \times (-d)}$

$=\frac{-1 \times d^{2 \times 3}}{d^4 \times (-1 \times d)}$

$=\frac{-1 \times d^6}{d^4 \times -1 \times d}$

$=\frac{-1 \times d^6}{-1 \times d^4 \times d}$

$=\frac{-1 \times d^6}{-1 \times d^{4+1}}$

$=\frac{-1 \times d^6}{-1 \times d^5}$

$=1 \times d^{6-5}$

$=1 \times d$

$=d$

Answer: $d$