Exercises: Power Laws

Simplify the following expressions with integer exponents as far as possible.

$(z^{2 k - 5} : z^{3}) : z^{k}$

For this task you need the following basic knowledge: Powers
Apply the power laws.
$(z^{2 k - 5} : z^{3}) : z^{k}$ $=$ $(z^{2 k - 5 - 3}) : z^{k}$
$=$ $z^{2 k - 8} : z^{k}$
$=$ $z^{2 k - 8 - k}$
$=$ $z^{k - 8}$
$90 \cdot 3^{n - 2} - 3^{n}$

For this task you need the following basic knowledge: Powers
Write $90$ as $3^{2} \cdot 10$ .
$90 \cdot 3^{n - 2} - 3^{n}$ $=$ $10 \cdot 3^{2} \cdot 3^{n - 2} - 3^{n}$
↓
Apply the power laws.
$=$ $10 \cdot 3^{2 + n - 2} - 3^{n}$
$=$ $10 \cdot 3^{n} - 3^{n}$
↓
Factorize $3^{n}$ .
$=$ $3^{n} \cdot (10 - 1)$
$=$ $9 \cdot 3^{n}$
↓
Write $9$ as $3^{2}$ .
$=$ $3^{2} \cdot 3^{n}$
↓
Apply the power laws.
$=$ $3^{2 + n}$
${[{(\frac{x}{4})}^{3}]}^{5} : {(\frac{x}{2})}^{6}$ for $x \neq 0$

For this task you need the following basic knowledge: Powers
Apply the power laws.
${[{(\frac{x}{4})}^{3}]}^{5} : {(\frac{x}{2})}^{6}$ $=$ ${(\frac{x}{4})}^{3 \cdot 5} : (\frac{x^{6}}{2^{6}})$
$=$ ${(\frac{x}{4})}^{15} : (\frac{x^{6}}{2^{6}})$
$=$ $(\frac{x^{15}}{4^{15}}) : (\frac{x^{6}}{2^{6}})$
$=$ $(\frac{x^{15}}{4^{15}}) \cdot (\frac{2^{6}}{x^{6}})$
↓
Write $2^{6}$ as $4^{3}$ , so you can shorten.
$=$ $(\frac{x^{15}}{4^{15}}) \cdot (\frac{4^{3}}{x^{6}})$
$=$ $\frac{x^{9}}{4^{12}}$

$\frac{{(3 a - 1)}^{2 k - 1}}{{(1 - 3 a)}^{2 k + 1}}$ for $a \neq \frac{1}{3}$

${(\frac{6 a^{2} b^{- 2}}{c^{n + 1} d^{2 n}})}^{3} : {[\frac{2 {(cd)}^{n}}{{ab}^{- 1}} \cdot \frac{c^{n} d^{2 n}}{3 {ab}^{- 2}}]}^{- 2}$ for $a, b, c, d \neq 0$

\frac{x^{2 a + 5}}{{(- y^{3})}^{2 b + 5} \cdot {[{(- z)}^{4}]}^{3 b + 3}} : \frac{x^{2 a}}{{(yz)}^{6 b + 10} \cdot {[{(- z)}^{3}]}^{2 b - 1}}

Assuming that $x, y, z > 0$ , $b \in ℤ$

${(\frac{2 a^{- 1} b^{2}}{3 {ac}^{- 2}})}^{- 3}$ for $a, b, c \neq 0$

For this task you need the following basic knowledge: Powers
Apply the power laws.
${(\frac{2 a^{- 1} b^{2}}{3 {ac}^{- 2}})}^{- 3}$ $=$ $\frac{2^{- 3} a^{3} b^{- 6}}{3^{- 3} a^{- 3} c^{6}}$
$=$ $\frac{a^{3} 3^{3} a^{3}}{2^{3} c^{6} b^{6}}$
$=$ $\frac{a^{3} \cdot 27 a^{3}}{8 c^{6} b^{6}}$
$=$ $\frac{27 a^{6}}{8 b^{6} c^{6}}$

${(\frac{u}{v})}^{n} \cdot {(\frac{v}{u})}^{3 n + 4} : {(\frac{- v}{u})}^{2 n + 1}$ for $u, v \neq 0$

$\frac{x^{5} + 1}{x^{m + 2}} - \frac{2 x^{2} - 2}{x^{m}} + \frac{2 - x}{x^{m - 2}}$ for $x \neq 0$

${(\frac{z - 3}{z + 5})}^{2 p + 1} \cdot {(\frac{5 + z}{z - 3})}^{p + 1} : {(\frac{z - 3}{z + 5})}^{4 p}$ for $z \in̸ {- 5; 3}$

${(1 + \frac{2}{t})}^{2} \cdot {[\frac{1}{t} - {(\frac{t}{2} - 1)}^{- 1}]}^{- 2}$ for $t \in̸ {- 2; 0; 2}$

Re-formulate the expression to a single fraction that does not contain any negative exponents.

$\frac{4 a^{- 1} z^{2}}{{(x^{2} y)}^{3}} : \frac{{(2 a)}^{- 3}}{{({xy}^{2} z)}^{- 2}}$

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$A$	$=$	$\frac{x^{2 a + 5}}{{(- y^{3})}^{2 b + 5} \cdot {[{(- z)}^{4}]}^{3 b + 3}} : \frac{x^{2 a}}{{(yz)}^{6 b + 10} \cdot {[{(- z)}^{3}]}^{2 b - 1}}$
	↓	Multiply out
	$=$	$\frac{x^{2 a + 5}}{{(- y^{3})}^{2 b + 5} \cdot {[{(- z)}^{4}]}^{3 b + 3}} \cdot \frac{{(yz)}^{6 b + 10} \cdot {[{(- z)}^{3}]}^{2 b - 1}}{x^{2 a}}$
	$=$	$\frac{x^{2 a + 5}}{{(- y^{3})}^{2 b + 5} \cdot {(- z)}^{4 (3 b + 3)}} \cdot \frac{{(yz)}^{6 b + 10} \cdot {(- z)}^{3 (2 b - 1)}}{x^{2 a}}$
	$=$	$\frac{x^{2 a + 5}}{{(- y)}^{6 b + 15} \cdot {(- z)}^{12 b + 12}} \cdot \frac{{(yz)}^{6 b + 10} \cdot {(- z)}^{6 b - 3}}{x^{2 a}}$
	↓	Factor out $(- 1)$ .
	$=$	$\frac{x^{2 a + 5}}{{(- 1)}^{6 b + 15} \cdot y^{6 b + 15} \cdot {(- 1)}^{12 b + 12} \cdot z^{12 b + 12}} \cdot \frac{{(yz)}^{6 b + 10} \cdot {(- 1)}^{6 b - 3} \cdot z^{6 b - 3}}{x^{2 a}}$
	↓	Decompose the $(y z)^{6 b + 10}$ .
	$=$	$\frac{x^{2 a + 5}}{{(- 1)}^{6 b + 15} \cdot y^{6 b + 15} \cdot {(- 1)}^{12 b + 12} \cdot z^{12 b + 12}} \cdot \frac{y^{6 b + 10} \cdot z^{6 b + 10} \cdot {(- 1)}^{6 b - 3} \cdot z^{6 b - 3}}{x^{2 a}}$
	$=$	$\frac{x^{2 a + 5 - 2 a}}{{(- 1)}^{6 b + 15} \cdot {(- 1)}^{12 b + 12}} \cdot y^{6 b + 10 - (6 b + 15)} \cdot z^{6 b + 10} \cdot {(- 1)}^{6 b - 3} \cdot z^{6 b - 3 - (12 b + 12)}$
	$=$	$\frac{x^{2 a + 5 - 2 a}}{{(- 1)}^{6 b + 15} \cdot {(- 1)}^{12 b + 12}} \cdot y^{6 b + 10 - 6 b - 15} \cdot z^{6 b + 10} \cdot {(- 1)}^{6 b - 3} \cdot z^{6 b - 3 - 12 b - 12}$
	↓	Resolve the bracket and simplify.
	$=$	$\frac{x^{5}}{{(- 1)}^{6 b + 15} \cdot {(- 1)}^{12 b + 12}} \cdot y^{- 5} \cdot z^{6 b + 10} \cdot {(- 1)}^{6 b - 3} \cdot z^{- 6 b - 15}$
	$=$	$\frac{x^{5} \cdot y^{- 5} \cdot z^{6 b + 10 - 6 b - 15} \cdot {(- 1)}^{6 b - 3}}{{(- 1)}^{6 b + 15} \cdot {(- 1)}^{12 b + 12}}$
	$=$	$\frac{x^{5} \cdot y^{- 5} \cdot z^{- 5} \cdot {(- 1)}^{6 b - 3}}{{(- 1)}^{6 b + 15} \cdot {(- 1)}^{12 b + 12}}$
	$=$	$\frac{x^{5} \cdot y^{- 5} \cdot z^{- 5} \cdot {(- 1)}^{6 b - 3}}{{(- 1)}^{6 b + 15 + 12 b + 12}}$
	$=$	$\frac{x^{5} \cdot y^{- 5} \cdot z^{- 5} \cdot {(- 1)}^{6 b - 3}}{{(- 1)}^{18 b + 27}}$
	↓	The factors of $(- 1)$ can be shortened.
	$=$	$\frac{x^{5} \cdot y^{- 5} \cdot z^{- 5} \cdot {(- 1)}^{6 b - 3 - (18 b + 27)}}{1}$
	$=$	$\frac{x^{5} \cdot y^{- 5} \cdot z^{- 5} \cdot {(- 1)}^{- 12 b - 30}}{1}$
	$=$	$x^{5} \cdot y^{- 5} \cdot z^{- 5} \cdot {(- 1)}^{- 12 b - 30}$
	$=$	$\frac{x^{5}}{y^{5} \cdot z^{5}} \cdot {(- 1)}^{- 12 b - 30}$

$(z^{2 k - 5} : z^{3}) : z^{k}$	$=$	$(z^{2 k - 5 - 3}) : z^{k}$
	$=$	$z^{2 k - 8} : z^{k}$
	$=$	$z^{2 k - 8 - k}$
	$=$	$z^{k - 8}$

$90 \cdot 3^{n - 2} - 3^{n}$	$=$	$10 \cdot 3^{2} \cdot 3^{n - 2} - 3^{n}$
	↓	Apply the power laws.
	$=$	$10 \cdot 3^{2 + n - 2} - 3^{n}$
	$=$	$10 \cdot 3^{n} - 3^{n}$
	↓	Factorize $3^{n}$ .
	$=$	$3^{n} \cdot (10 - 1)$
	$=$	$9 \cdot 3^{n}$
	↓	Write $9$ as $3^{2}$ .
	$=$	$3^{2} \cdot 3^{n}$
	↓	Apply the power laws.
	$=$	$3^{2 + n}$

${[{(\frac{x}{4})}^{3}]}^{5} : {(\frac{x}{2})}^{6}$	$=$	${(\frac{x}{4})}^{3 \cdot 5} : (\frac{x^{6}}{2^{6}})$
	$=$	${(\frac{x}{4})}^{15} : (\frac{x^{6}}{2^{6}})$
	$=$	$(\frac{x^{15}}{4^{15}}) : (\frac{x^{6}}{2^{6}})$
	$=$	$(\frac{x^{15}}{4^{15}}) \cdot (\frac{2^{6}}{x^{6}})$
	↓	Write $2^{6}$ as $4^{3}$ , so you can shorten.
	$=$	$(\frac{x^{15}}{4^{15}}) \cdot (\frac{4^{3}}{x^{6}})$
	$=$	$\frac{x^{9}}{4^{12}}$

$\frac{{(3 a - 1)}^{2 k - 1}}{{(1 - 3 a)}^{2 k + 1}}$	$=$	$\frac{{(- 1)}^{2 k - 1} {(- 3 a + 1)}^{2 k - 1}}{{(- 3 a + 1)}^{2 k + 1}}$
	↓	Divide and apply the power laws.
	$=$	${(- 1)}^{2 k - 1} {(- 3 a + 1)}^{2 k - 1 - (2 k + 1)}$
	↓	Multiply out.
	$=$	${(- 1)}^{2 k - 1} {(- 3 a + 1)}^{2 k - 1 - 2 k - 1}$
	$=$	${(- 1)}^{2 k - 1} {(- 3 a + 1)}^{- 2}$
	↓	A negative exponent corresponds to a fraction.
	$=$	${(- 1)}^{2 k - 1} \cdot \frac{1}{{(- 3 a + 1)}^{2}}$
	$=$	$\frac{{(- 1)}^{2 k - 1}}{{(- 3 a + 1)}^{2}}$
	$=$	$- \frac{1}{(- 3 a + 1)^{2}}$

${(\frac{6 a^{2} b^{- 2}}{c^{n + 1} d^{2 n}})}^{3} : {[\frac{2 {(cd)}^{n}}{{ab}^{- 1}} \cdot \frac{c^{n} d^{2 n}}{3 {ab}^{- 2}}]}^{- 2}$	$=$	$(\frac{6^{3} a^{6} b^{- 6}}{c^{3 n + 3} d^{6 n}}) : {(\frac{{ab}^{- 1} \cdot 3 {ab}^{- 2}}{2 {(cd)}^{n} \cdot c^{n} d^{2 n}})}^{2}$
	$=$	$(\frac{6^{3} a^{6}}{c^{3 n + 3} d^{6 n} \cdot b^{6}}) : (\frac{a^{2} b^{- 2} \cdot 3^{2} a^{2} b^{- 4}}{2^{2} {(cd)}^{2 n} \cdot c^{2 n} d^{4 n}})$
	↓	Convert division into multiplication by "flipping" the fraction.
	$=$	$(\frac{6^{3} a^{6}}{c^{3 n + 3} d^{6 n} \cdot b^{6}}) \cdot (\frac{2^{2} {(cd)}^{2 n} \cdot c^{2 n} d^{4 n}}{a^{2} b^{- 2} \cdot 3^{2} a^{2} b^{- 4}})$
	↓	Conclude.
	$=$	$\frac{216 a^{6} 4 c^{4 n} d^{6 n}}{9 a^{4} c^{3 n + 3} d^{6 n}}$
	$=$	$\frac{216 a^{2} 4 c^{n - 3}}{9}$
	$=$	$96 a^{2} c^{n - 3}$

${(\frac{2 a^{- 1} b^{2}}{3 {ac}^{- 2}})}^{- 3}$	$=$	$\frac{2^{- 3} a^{3} b^{- 6}}{3^{- 3} a^{- 3} c^{6}}$
	$=$	$\frac{a^{3} 3^{3} a^{3}}{2^{3} c^{6} b^{6}}$
	$=$	$\frac{a^{3} \cdot 27 a^{3}}{8 c^{6} b^{6}}$
	$=$	$\frac{27 a^{6}}{8 b^{6} c^{6}}$

${(\frac{u}{v})}^{n} \cdot {(\frac{v}{u})}^{3 n + 4} : {(\frac{- v}{u})}^{2 n + 1}$	$=$	$\frac{u^{n}}{v^{n}} \cdot \frac{v^{3 n + 4}}{u^{3 n + 4}} : \frac{- v^{2 n + 1}}{u^{2 n + 1}}$
	↓	Write the division as a fraction.
	$=$	$\frac{\frac{u^{n}}{v^{n}} \cdot \frac{v^{3 n + 4}}{u^{3 n + 4}}}{\frac{- v^{2 n + 1}}{u^{2 n + 1}}}$
	↓	Resolve the double fraction.
	$=$	$\frac{u^{n}}{v^{n}} \cdot \frac{v^{3 n + 4}}{u^{3 n + 4}} \cdot \frac{u^{2 n + 1}}{- v^{2 n + 1}}$
	$=$	$\frac{u^{n} \cdot v^{3 n + 4} \cdot u^{2 n + 1}}{v^{n} \cdot u^{3 n + 4} \cdot (- v^{2 n + 1})}$
	$=$	$\frac{u^{n + 2 n + 1} \cdot v^{3 n + 4}}{u^{3 n + 4} \cdot (- 1) \cdot v^{n + 2 n + 1}}$
	$=$	$u^{3 n + 1 - (3 n + 4)} \cdot (- 1) \cdot v^{3 n + 4 - (3 n + 1)}$
	↓	Shorten by using the power laws.
	$=$	$u^{- 3} \cdot (- 1) \cdot v^{3}$
	$=$	$\frac{(- 1) \cdot v^{3}}{u^{3}}$
	$=$	$\frac{- v^{3}}{u^{3}}$
	$=$	$- {(\frac{v}{u})}^{3}$

$\frac{x^{5} + 1}{x^{m + 2}} - \frac{2 x^{2} - 2}{x^{m}} + \frac{2 - x}{x^{m - 2}}$	$=$	$\frac{x^{5} + 1}{x^{m + 2}} - \frac{2 x^{4} - 2 x^{2}}{x^{m + 2}} + \frac{2 - x}{x^{m} \cdot x^{- 2}}$
	↓	Apply the power laws to $x^{- 2}$ .
	$=$	$\frac{x^{5} + 1}{x^{m + 2}} - \frac{2 x^{4} - 2 x^{2}}{x^{m + 2}} + \frac{2 x^{2} - x^{3}}{x^{m}}$
	↓	Get to a common denominator.
	$=$	$\frac{x^{5} + 1}{x^{m + 2}} - \frac{2 x^{4} - 2 x^{2}}{x^{m + 2}} + \frac{2 x^{4} - x^{5}}{x^{m + 2}}$
	$=$	$\frac{x^{5} + 1 - 2 x^{4} + 2 x^{2} + 2 x^{4} - x^{5}}{x^{m + 2}}$
	$=$	$\frac{1 + 2 x^{2}}{x^{m + 2}}$

${(\frac{z - 3}{z + 5})}^{2 p + 1} \cdot {(\frac{5 + z}{z - 3})}^{p + 1} : {(\frac{z - 3}{z + 5})}^{4 p}$	$=$	${(\frac{z - 3}{z + 5})}^{2 p + 1 - 4 p} \cdot {(\frac{5 + z}{z - 3})}^{p + 1}$
	$=$	${(\frac{z - 3}{z + 5})}^{1 - 2 p} \cdot {(\frac{5 + z}{z - 3})}^{p + 1}$
	$=$	${(\frac{z - 3}{z + 5})}^{1 - 2 p} \cdot {(\frac{z - 3}{z + 5})}^{- (p + 1)}$
	↓	Resolve the bracket.
	$=$	${(\frac{z - 3}{z + 5})}^{1 - 2 p} \cdot {(\frac{z - 3}{z + 5})}^{- p - 1}$
	↓	Apply the power laws.
	$=$	${(\frac{z - 3}{z + 5})}^{1 - 2 p + (- p) - 1}$
	$=$	${(\frac{z - 3}{z + 5})}^{- 3 p}$
	$=$	${(\frac{z + 5}{z - 3})}^{3 p}$

$\frac{4 a^{- 1} z^{2}}{{(x^{2} y)}^{3}} : \frac{{(2 a)}^{- 3}}{{({xy}^{2} z)}^{- 2}}$	$=$	$\frac{4 \cdot \frac{1}{a} \cdot z^{2}}{x^{6} \cdot y^{3}} : \frac{\frac{1}{{(2 a)}^{3}}}{\frac{1}{{({xy}^{2} z)}^{2}}}$
	↓	Division can be done by "flipping the fraction".
	$=$	$\frac{4 \cdot \frac{1}{a} \cdot z^{2}}{x^{6} \cdot y^{3}} \cdot \frac{\frac{1}{{({xy}^{2} z)}^{- 2}}}{\frac{1}{{(2 a)}^{3}}}$
	↓	Resolve the double fraction.
	$=$	$\frac{4 \cdot \frac{1}{a} \cdot z^{2}}{x^{6} \cdot y^{3}} \cdot \frac{1}{{({xy}^{2} z)}^{2}} \cdot \frac{{(2 a)}^{3}}{1}$
	$=$	$\frac{\frac{4 z^{2}}{a}}{x^{6} \cdot y^{3}} \cdot \frac{8 a^{3}}{x^{2} \cdot y^{4} \cdot z^{2}}$
	$=$	$\frac{\frac{4 z^{2} \cdot 8 a^{3}}{a}}{x^{6 + 2} \cdot y^{3 + 4} \cdot z^{2}}$
	↓	Finally, shorten.
	$=$	$\frac{4 \cdot 8 a^{2}}{x^{8} y^{7}}$
	$=$	$\frac{32 a^{2}}{x^{8} y^{7}}$