Exponents and Radicals: Problems with Solutions

Solution:
$7^{2}<3^{4}$ because $49<81$

Solution:
$4^{2}=2^{4}$ because $16=16$

Solution:
$3^{5}>5^{3}$ because $243>125$

Solution:
Property 1 is used.
$x^{n} \cdot x^{m}=x^{n+m}\Longrightarrow 3^{5}\cdot 3^{2} \cdot 3^{-3}=3^{5+2-3}=3^{4}$

Solution:
Property 2 is used.
$\frac{x^{n}}{x^{m}}=x^{n-m}$

$\frac{2^{12}}{2^{8}}=2^{12-8}=2^{4}=16$

Solution:
Property 3 is used.
$(x^{n})^{m}=x^{n\cdot m}$

$(2^{2})^{3}=2^{6}=64$

Solution:
Property 5 is used.
$(\frac{x}{y})^{n}=\frac{x^{n}}{y^{n}}$

$\left( \frac{5}{2}\right) ^{3}=\frac{5^{3}}{2^{3}}=\frac{125}{8}$

Solution:
If we use property 1, we have
$5^{2} \cdot 5^{3}=5^{2+3}=5^{5}$.

Solution:
Property 4 is used.
$(x \cdot y)^{n}=x^{n}\cdot y^{n}$ $(2\times 3)^{3}=2^{3}\times 3^{3}=8\times 27=216$

Solution:
If we use property 1, we have
$3^{10}=3^{6} \cdot 3^{x}=3^{6+x}$
$3^{10}=3^{6+x}$
$10=6+x$
$x=4$

Solution:
$a^{0}\cdot a^{8}=a^{0+8}=a^{8}$

Solution:
$(a^{5})^{6}=a^{5\cdot 6}=a^{30}$

Solution:
$(3^{2})^{3}=3^{6}=n^{6}\Longrightarrow n=3$

Solution:
$\frac{4^{8}}{4^{4}}=4^{8-4}=4^{4}$

Solution:
If we use property 2, we have
$\frac{a\cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a\cdot a}=\frac{a^{9}}{a^{2}}=a^{7}$

Solution:
$\frac{5^{6}}{5^{-1}}=5^{6-(-1)}=5^{7}$

Solution:
$\frac{x^{12}}{x^{6}}=x^{12-6}=x^{6}$

Solution:
$\frac{9^{x}}{9^{2}}=9^{x-2}=9^{5}\Longrightarrow x-2=5\Longrightarrow x=7$

Solution:
$2^{4}(x^{3})^{4}=16x^{12}$

Solution:
If we use property 5, we have
$(\frac{z^{2}}{2})^{3}=\frac{(z^{2})^{3}}{2^{3}}=\frac{z^{6}}{8}$

Solution:
$(2\cdot y^{4})^{3}=2^{3}\cdot (y^{4})^{3}=8y^{12}$.

Solution:
$x^{1+\frac{5}{3}}=x^{\frac{3+5}{3}}=x^{\frac{8}{3}}=\sqrt[3]{x^{8}}$

Solution:
$\left( \sqrt[3]{x}\right) ^{2}=\left( x^{\frac{1}{3}}\right)^{2}=x^{\frac{2}{3}}$

Solution:
$x^{1-\frac{3}{2}+\frac{5}{4}}=x^{\frac{4-6+5}{4}}=x^{\frac{3}{4}}=\sqrt[4]{x^{3}}$

Solution:
$\left( \sqrt{\sqrt[3]{x}}\right)^{5}=\left( \left( x^{\frac{1}{3}}\right) ^{\frac{1}{2}}\right) ^{5}=\left( x^{\frac{1}{6}}\right)^{5}=x^{\frac{5}{6}}$

Solution:
$\left( \frac{x^{2}}{2y}\right) ^{3}-\left( \frac{x^{2}}{y^{3}}\right) =\frac{x^{6}}{2^{3}y^{3}}-\frac{x^{2}}{y^{3}}=\frac{x^{6}-8x^{2}}{8y^{3}}$

Solution:
$\left( \frac{x^{2}}{2y}\div \frac{2y^{2}}{3x}\right)^{2}=\left(\frac{x^{2}(3x)}{2y^{2}(2y)}\right)^{2}=\left( \frac{3x^{3}}{4y^{3}}\right)^{2}=\frac{9x^{6}}{16y^{6}}$

Solution:
$\left( \frac{a \times b}{2c}\right)^{3}\div \left( \frac{a \cdot b}{c^{3}}\right) ^{2}=\frac{a^{3}\cdot b^{3}}{2^{3}c^{3}}\div \frac{a^{2} \cdot b^{2}}{c^{6}}=\frac{a^{3} \cdot b^{3} \cdot c^{6}}{a^{2}\cdot b^{2} \cdot 2^{3}c^{3}}=\frac{a^{3-2} \cdot b^{3-2} \cdot c^{6-3}}{8}=\frac{abc^{3}}{8}$

Solution:
If $x=3$ then
$5^{x}-3^{x}=5^{3}-3^{3}=125-27=98$

Solution:
$3^{n}=3^{2}\cdot 3^{3}=3^{2+3}=3^{5}$
Therefore, $3^{n}=3^{5}$
So $n=5$

Solution:
$8^{n}=2^{-2} \cdot 4^{3}\Longrightarrow (2^{3})^{n}=2^{-2} \cdot (2^{2})^{3}\Longrightarrow 2^{3n}=2^{6-2}\Longrightarrow 2^{3n}=2^{4}$

$3n=4$, so $n=\frac{4}{3}$

Solution:
$2^{2x-1}=4\Longrightarrow 2^{2x-1}=2^{2}\Longrightarrow 2x-1=2\Longrightarrow x=\frac{3}{2}$

Solution:
$27^{n}=3^{2} \cdot 9^{3}$
$(3^{3})^{n}=3^{2} \cdot(3^{2})^{3}$
$3^{3n}=3^{8}$
$3n=8$
$n=\frac{8}{3}$

Solution:
$\sqrt[3]{8^{x}}=128$
$\sqrt[3]{(2^{3})^{x}}=2^{7}$
$2^{\frac{3x}{3}}=2^{7}\Longrightarrow x=7$

Solution:
[tex]\left(\dfrac{1}{3}\right)^{-1}=3, \left(-\dfrac{1}{3}\right)^{-2}=(-3)^2=9[/tex]
[tex]3-9=-6[/tex]

Solution:
[tex]2^{30}+2^{30}+2^{30}+2^{30}=4\times 2^{30}=2^2\times 2^{30}=2^{32}[/tex]

Solution:
[tex]32^{\frac{1}{5}}=2, \left(8^{\frac{2}{3}}\right)^{\frac{1}{2}}=8^{\frac{1}{3}}=2, \left(\dfrac{1}{27}\right)^{-\frac{1}{3}}=3, 4^{-\frac{1}{2}}=\dfrac{1}{2}[/tex]
[tex](2+2)\times \left(3-\dfrac{1}{2}\right)=4\times \dfrac{5}{2}=10[/tex]

Solution:
[tex]s=2^{\frac{1}{2}}=\sqrt{2}:\ (s-3)^2+2s(3-s)=(3-s)(3-s+2s)=(3-s)(3+s)=9-2=7[/tex]
[tex]7+6=13[/tex]

Solution:
[tex]\left(3^{\frac{1}{2}}\right)^6=3^3=27, \left(5^{\frac{1}{3}}\right)^6=5^2=25[/tex]
[tex]27\gt 25 \Rightarrow 3^{\frac{1}{2}}\gt 5^{\frac{1}{3}}[/tex]

Solution:
[tex]2^{2006}\times 2^{2008}=2^{4014}, 3\times \left(2^{2007}\right)^2=3\times 2^{4014}[/tex]
[tex]\dfrac{2^{4014}\times 2007}{3\times 2^{4014}}=\dfrac{2007}{3}=669[/tex]