[tex]\boxed{\begin{array}{c|c}a^m\cdot a^n=a^{m+n}&a^m:a^n=a^{m-n}\\a^m\cdot b^m=(a\cdot b)^m&a^m:b^m=(a:b)^m\\a^0=1&a^1=a\\a^{-1}=\dfrac1{a}&a^{-n}=\dfrac{1}{a^n}\\a^{\frac{n}{2}}=\sqrt{a^n}&a^{\frac{n}{m}}=\sqrt[m]{a^n}\\(a^n)^m=a^{n\cdot m}\end{array}}[/tex]
a)
[tex]4^{-3}=\dfrac1{4^3}=\dfrac1{64}[/tex]
b)
[tex]\left(\dfrac34\right)^2=\dfrac9{16}[/tex]
c)
d)
[tex]\dfrac{2^3\cdot 2^5}{2^4}=\dfrac{2^8}{2^4}=2^4=16[/tex]
e)
[tex]\dfrac{5^4\cdot 5^{-2}}{5^3}=\dfrac{5^2}{5^3}=5^{-1}=\dfrac15[/tex]
f)
[tex]9^{\frac13}:9^{-\frac16}=9^{\frac13+\frac16}=9^{\frac26+\frac16}=9^{\frac36}=9^{\frac12}=\sqrt9=3[/tex]
g)
[tex]2^{\frac25}\cdot8^{\frac15}=2^{\frac25}\cdot(2^3)^{\frac15}=2^{\frac25}\cdot 2^{\frac35}=2^{\frac55}=2^1=2[/tex]
h)
[tex]125^{\frac43}=(5^3)^{\frac43}=5^4=625[/tex]
i)
[tex]\sqrt[5]{32^4}=32^{\frac45}=(2^5)^{\frac45}=2^4=16[/tex]
j)
[tex]\left(\sqrt7^{\sqrt2}\right)^{\sqrt2}=\left[\left(7^{\frac12}\right)^{2^{\frac12}}\right]^{2^{\frac12}}=7^{\frac12\cdot 2^{\frac12}\cdot 2^{\frac12}}=7^{2^{-1+\frac12+\frac12}}=7^{2^{0}}=7^{1}=7[/tex]
k)
[tex]\dfrac{6^{\sqrt3+1}\cdot 2^{-\sqrt3}}{3^{\sqrt3}}=\dfrac{(2\cdot 3)^{\sqrt3+1}\cdot 2^{-\sqrt3}}{3^{\sqrt3}}=\dfrac{2^{\sqrt3+1-\sqrt3}\cdot 3^{\sqrt3+1}}{3^{\sqrt3}}=\dfrac{2^1\cdot3^{\sqrt3+1}}{3^\sqrt3}=2\cdot 3=6[/tex]
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Działania na potęgach
[tex]\boxed{\begin{array}{c|c}a^m\cdot a^n=a^{m+n}&a^m:a^n=a^{m-n}\\a^m\cdot b^m=(a\cdot b)^m&a^m:b^m=(a:b)^m\\a^0=1&a^1=a\\a^{-1}=\dfrac1{a}&a^{-n}=\dfrac{1}{a^n}\\a^{\frac{n}{2}}=\sqrt{a^n}&a^{\frac{n}{m}}=\sqrt[m]{a^n}\\(a^n)^m=a^{n\cdot m}\end{array}}[/tex]
Rozwiazanie:
a)
[tex]4^{-3}=\dfrac1{4^3}=\dfrac1{64}[/tex]
b)
[tex]\left(\dfrac34\right)^2=\dfrac9{16}[/tex]
c)
[tex]\left(\dfrac34\right)^2=\dfrac9{16}[/tex]
d)
[tex]\dfrac{2^3\cdot 2^5}{2^4}=\dfrac{2^8}{2^4}=2^4=16[/tex]
e)
[tex]\dfrac{5^4\cdot 5^{-2}}{5^3}=\dfrac{5^2}{5^3}=5^{-1}=\dfrac15[/tex]
f)
[tex]9^{\frac13}:9^{-\frac16}=9^{\frac13+\frac16}=9^{\frac26+\frac16}=9^{\frac36}=9^{\frac12}=\sqrt9=3[/tex]
g)
[tex]2^{\frac25}\cdot8^{\frac15}=2^{\frac25}\cdot(2^3)^{\frac15}=2^{\frac25}\cdot 2^{\frac35}=2^{\frac55}=2^1=2[/tex]
h)
[tex]125^{\frac43}=(5^3)^{\frac43}=5^4=625[/tex]
i)
[tex]\sqrt[5]{32^4}=32^{\frac45}=(2^5)^{\frac45}=2^4=16[/tex]
j)
[tex]\left(\sqrt7^{\sqrt2}\right)^{\sqrt2}=\left[\left(7^{\frac12}\right)^{2^{\frac12}}\right]^{2^{\frac12}}=7^{\frac12\cdot 2^{\frac12}\cdot 2^{\frac12}}=7^{2^{-1+\frac12+\frac12}}=7^{2^{0}}=7^{1}=7[/tex]
k)
[tex]\dfrac{6^{\sqrt3+1}\cdot 2^{-\sqrt3}}{3^{\sqrt3}}=\dfrac{(2\cdot 3)^{\sqrt3+1}\cdot 2^{-\sqrt3}}{3^{\sqrt3}}=\dfrac{2^{\sqrt3+1-\sqrt3}\cdot 3^{\sqrt3+1}}{3^{\sqrt3}}=\dfrac{2^1\cdot3^{\sqrt3+1}}{3^\sqrt3}=2\cdot 3=6[/tex]