4.4

1. Simplify. Express each answer with positive exponents.

a)
$x^4(x^3)=x^{4+3}=x^7$

b)
$(p^{-3})(p^5)=p^{-3+5}=p^2$

c)
$\frac{m^5}{m^{-3}}=m^{5-(-3)}=m^8$

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

e)
$(y^3{)}^2=y^{3\cdot 2}=y^6$

f)
$(k^6{)}^{-2}=k^{6\cdot (-2)}=k^{-12}=\frac{1}{k^{12}}$

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2. Simplify. Express each answer with positive exponents.

a)
$y^{10}(y^4{)}^{-3}=y^{10}\cdot y^{-12}=y^{-2}=\frac{1}{y^2}$

b)
$\frac{x^9}{x^{-5}}=x^{9-(-5)}=x^{14}$

c)
$\frac{(n^{-4}{)}^3}{(n^{-3}{)}^{-4}}=\frac{n^{-12}}{n^{12}}=n^{-24}=\frac{1}{n^{24}}$

d)
$\frac{b^{-14}\cdot b^9}{b^5}=\frac{b^{-5}}{b^5}=b^{-10}=\frac{1}{b^{10}}$

e)
$\frac{(w^{-2}x{)}^2}{(x^{-1}{)}^2{w}^3}=\frac{w^{-4}{x}^2}{x^{-2}{w}^3}=w^{-7}{x}^4=\frac{x^4}{w^7}$

f)
$(p{q}^2{)}^{-1}(p^3{q}^3)=p^{-1}{q}^{-2}\cdot p^3{q}^3=p^{2}q$

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3. Evaluate the expression for (x = 2) and (y = 3).

a)
Substitute directly:
$\frac{2^7\cdot (3^2{)}^3}{2^5\cdot 3^4}=\frac{128\cdot 729}{32\cdot 81}=\frac{93312}{2592}=36$

b)
Simplify first:
$\frac{x^7{y}^6}{x^5{y}^4}=x^2{y}^2=(2{)}^2(3{)}^2=4\cdot 9=36$

c)
Simplifying first is more efficient as it reduces computation steps.

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4. Simplify. Express answers with positive exponents.

a)
$\frac{(x^{3}y{)}^{-1}(x^4{y}^3)}{(x^2{y}^{-3}{)}^{-2}}=\frac{x^{-3}{y}^{-1}\cdot x^4{y}^3}{x^{-4}{y}^6}=\frac{x^1{y}^2}{x^{-4}{y}^6}=x^5{y}^{-4}=\frac{x^5}{y^4}$

b)
$\frac{(3m^4{n}^2{)}^2}{12m^{-2}{n}^6}=\frac{9m^8{n}^4}{12m^{-2}{n}^6}=\frac{3}{4}{m}^{10}{n}^{-2}=\frac{3m^{10}}{4n^2}$

c)
$\frac{(2a^3{)}^2}{4a{b}^2}=\frac{4a^6}{4a{b}^2}=\frac{a^5}{b^2}$

d)
$\frac{(w^{-2}x{)}^2}{(x^{-1}{)}^2{w}^3}=\frac{w^{-4}{x}^2}{x^{-2}{w}^3}=w^{-7}{x}^4=\frac{x^4}{w^7}$

e)
$\frac{(ab{)}^{-2}}{b^5}=\frac{a^{-2}{b}^{-2}}{b^5}=a^{-2}{b}^{-7}=\frac{1}{a^2{b}^7}$

f)
$(p{q}^2{)}^{-1}(p^3{q}^3)=p^{-1}{q}^{-2}\cdot p^3{q}^3=p^{2}q$

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5. Simplify. Express answers with positive exponents.

a)
$\frac{(x^{-3}{)}^{-3}(x^{-1}{)}^5}{(x^{-1}{)}^{4}x}=\frac{x^9{x}^{-5}}{x^{-4}x}=\frac{x^4}{x^{-3}}=x^7$

b)
$\frac{(p^{-5}{r}^3{)}^2}{(p^{2}r{)}^2(p^{-1}{)}^2}=\frac{p^{-10}{r}^6}{p^2{r}^2{p}^{-2}}=\frac{p^{-10}{r}^6}{p^0{r}^2}=p^{-10}{r}^4=\frac{r^4}{p^{10}}$

c)
$\frac{(10x{)}^{-1}{y}^3}{15x^3{y}^{-3}}=\frac{y^3/(10x)}{15x^3/y^3}=\frac{y^6}{150x^4}$

d)
$(3x{y}^4{)}^2(2x^{2}y{)}^3=9x^2{y}^8\cdot 8x^6{y}^3=72x^8{y}^{11}$

e)
$\frac{a(ab{)}^{-1}}{(a^2{b}^{-3}{)}^{-2}}=\frac{a\cdot a^{-1}{b}^{-1}}{a^{-4}{b}^6}=\frac{b^{-1}}{a^{-4}{b}^6}=a^4{b}^{-7}=\frac{a^4}{b^7}$

f)
$\frac{m^2{n}^2}{(m^3{n}^{-2}{)}^{-2}}=\frac{m^2{n}^2}{m^{-6}{n}^4}=m^8{n}^{-2}=\frac{m^8}{n^2}$

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6. Simplify. Express answers in rational form with positive exponents.

a)
$\sqrt{1024x^{20}}=32x^{10}$

b)
$\sqrt[9]{512x^{27}}=2x^3$

c)
$\frac{\sqrt[3]{(10x^3{)}^2}}{(10x^6{)}^{-1}}=\frac{10^{2/3}{x}^2}{1/(10x^6)}=10^{5/3}{x}^8$

d)
$9(c^8{)}^{0.5}=9c^4$

e)
$(16c^{12}{)}^{0.25}=2c^3$

f)
${\left[(32x^5{)}^{-2}(x^{-1}{)}^{10}\right]}^{0.2}={\left[2^{-10}{x}^{-20}\right]}^{0.2}=2^{-2}{x}^{-4}=\frac{1}{4x^4}$

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7. Evaluate. Express answers in rational form with positive exponents.

a)
$(16x^6{y}^4{)}^{1/2}\text{ for }x=2,y=5:\sqrt{16\cdot 64\cdot 625}=\sqrt{640000}=800$

b)
$8(x^9{y}^3{)}^{1/3}\text{ for }x=10,y=1:8\cdot 10^3=8000$

c)
$(9p^{-2}{)}^{1/(6p^2)}\text{ for }p=3:(9\cdot 3^{-2}{)}^{1/54}=(1{)}^{1/54}=1$

d)
$(81x^4{y}^6{)}^{1/2}\text{ for }x=1,y=2:\sqrt{81\cdot 1\cdot 64}=\sqrt{5184}=72$

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8. Evaluate. Express answers in rational form with positive exponents.

a)
$(10000x{)}^3\text{ for }x=5:(50000{)}^3=125000000000$

b)
${\left(64a^{12}(a^{1.5}{)}^{-6}\right)}^{0.5}\text{ for }a=1:\sqrt{64\cdot 1}=8$

c)
$(9m^4{n}^6{)}^{0.5}(81m^{12}{n}^8{)}^{0.25}\text{ for }m=10,n=1:9\cdot 3\cdot 10^5=900000$

d)
$\frac{\sqrt{(18m^{-5}{n}^{-2})(32m^{2}n)}}{4m{n}^{-3}}\text{ for }m=10,n=1:\frac{24\cdot 10^{-1.5}}{40}=\frac{6}{10^{2.5}}=\frac{3}{50\sqrt{10}}$

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9. Simplify. Express answers in rational form with positive exponents.

a)
${\left[\frac{(x^{18}{)}^{-1}}{6}\cdot \sqrt[3]{-243x^{10}}\right]}^{0.5}={\left[\frac{-3x^{-8}}{6}\right]}^{0.5}={\left[-\frac{x^{-8}}{2}\right]}^{0.5}\text{ (Not real)}$

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11. Cylinder Surface Area-to-Volume Ratio

a)
$\frac{SA}{V}=\frac{2\pi r(r+h)}{\pi r^{2}h}=\frac{2(r+h)}{rh}$

b)
For (r = 0.8), (h = 12):
$\frac{2(0.8+12)}{0.8\cdot 12}=\frac{25.6}{9.6}\approx 2.667$

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12. Order Expressions

For (x = 22), (y = 3):

1. $\frac{1}{x^6{y}^7}\approx 4.05\times 10^{-12}$

2. $\frac{1}{x^5{y}^8}\approx 2.96\times 10^{-11}$

3. $\frac{y^4}{x^5}\approx 0.0000157$

   Order: $\frac{1}{x^6{y}^7}<\frac{1}{x^5{y}^8}<\frac{y^4}{x^5}$

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13. Simplifying Algebraic vs. Numerical Expressions

Similarity: Both use exponent rules and order of operations.
Difference: Algebraic expressions retain variables, while numerical expressions resolve to a single number.