Esitä murtopotenssi juurimuodossa ja sievennä
a) \(16^{\frac{1}{4}}\)
b) \(7^{\frac{2}{3}}\)
c) \(8^{\small{-}\frac{2}{3}}\)
d) \(x^{\frac{9}{4}}\)
e) \(y^{\frac{4}{3}}\)
f) \(a^{-\frac{1}{4}}\)
Merkitse vihkoosi värikynällä pisteesi ja puuttuvat välivaiheet.
\(\begin{align*}\textbf{a) }\quad & 16^{\frac{1}{4}}\\=&\ \sqrt[4]{16}=\underline{\underline{\ 2\ }} \quad \require{color}\color{red}{\text{(+1p)}}\end{align*}\) \(\)
\(\begin{align*}\textbf{b) }\quad & 7^{\frac{2}{3}}\\=&\sqrt[3]{7^2}=\underline{\underline{\sqrt[3]{49}}} \quad \require{color}\color{red}{\text{(+1p)}}\end{align*}\)
\(\begin{align*}\textbf{c)}\quad &8^{-\frac{2}{3}}=\frac{1}{8^\frac{2}{3}}\\ =&\frac{1}{\sqrt[3]{8^2}}=\frac{1}{\sqrt[3]{(2^3)^2}}\\ =& \frac{1}{\sqrt[3]{2^6}}= \frac{1}{2^\frac{6}{3}}\\ =&\frac{1}{2^2}=\underline{\underline{\frac{1}{4}}} \qquad \require{color}\color{red}{\text{(+1p)}}\end{align*}\)
\(\begin{align*}\textbf{d)}\quad x^{\frac{9}{4}}=&\ \sqrt[4]{x^9}\\ =&\underline{\underline{x^2\sqrt[4]{x}}} \quad \require{color}\color{red}{\text{(+1p)}}\end{align*}\)
\(\begin{align*}\textbf{e)}\quad &y^{\frac{4}{3}}\\=&\sqrt[3]{y^4}=\underline{\underline{y\sqrt[3]{y}}} \quad \require{color}\color{red}{\text{(+1p)}}\end{align*}\)
\(\begin{align*}\textbf{f)}\quad &a^{-\frac{1}{4}}\\=&\sqrt[4]{a^{-1}}\\=&\sqrt[4]{\frac{1}{a}}\\=&\frac{\sqrt[4]{1}}{\sqrt[4]{a}}=\underline{\underline{\frac{1}{ \sqrt[4]{a}}}} \quad \require{color}\color{red}{\text{(+1p)}}\end{align*}\)
Esitä juuri murtopotenssimuodossa ja sievennä
a) \(\sqrt{3}\)
b) \(\sqrt[3]{11}\)
c) \(\sqrt{\sqrt{81}}\)
d) \(\sqrt[22]{2^{21}}\)
e) \(y^3 \sqrt{y}\)
f) \(\dfrac{1}{\sqrt[4]{a^3}}\)
Merkitse vihkoosi värikynällä pisteesi ja puuttuvat välivaiheet.
\(\require{color} \textbf{a) }\sqrt{3}=\underline{\underline{3^{\frac{1}{2}}}} \qquad \color{red}{\text{(+1p)}}\)
\(\require{color} \textbf{b) }\sqrt[3]{11}=\underline{\underline{11^{\frac{1}{3}}}} \qquad \color{red}{\text{(+1p)}}\)
\(\require{color} \begin{align*}\textbf{c)}\quad &\sqrt[]{\sqrt{81}}=\Big(81^{\frac{1}{2}}\Big)^{\frac{1}{2}}\\\\=& {3^{4 \cdot \frac{1}{2} \cdot \frac{1}{2}}}=\underline{\underline{3 }}\quad \color{red}{\text{(+1p)}}\end{align*}\)
\( \begin{align*} \textbf{d) } \sqrt[22]{2^{21}}&=(2^{21})^{\frac{1}{22}} =\underline{\underline{2^{\frac{21}{22}}}} \require{color} \quad \color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*} \textbf{e) }y^3 \sqrt[]{y} &= y^3 \cdot y^{\frac{1}{2}} =\underline{\underline{y^\frac{7}{2}}} \quad \require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\require{color} \begin{align*}\textbf{f)}\quad &\frac{1}{ \sqrt[4]{a^3}}=\frac{1}{a^{\frac{3}{4}}}=\underline{\underline{a^{-\frac{3}{4}}}} \quad \color{red}{\text{(+1p)}}\end{align*}\)
a) Laske \(\sqrt{3} \cdot \sqrt[4]{9} \cdot \sqrt[3]{27}\)
b) Ratkaise yhtälö \(x^{\frac{1}{9}} \cdot x^{\frac{2}{9}}=3\)
Merkitse vihkoosi värikynällä pisteesi ja puuttuvat välivaiheet.
\(\begin{align*}\textbf{a)}\quad &\sqrt[]{3} \cdot \sqrt[4]{9}\cdot \sqrt[3]{27} \\ =&\ 3^{\frac{1}{2}} \cdot 9^{\frac{1}{4}}\cdot 27^{\frac{1}{3}} \quad && \require{color}\color{red}{\text{(+1p)}}\\ =&\ 3^{\frac{1}{2}} \cdot (3^2)^{\frac{1}{4}}\cdot (3^3)^{\frac{1}{3}}\\=&\ 3^{\frac{1}{2}}\cdot 3^{\frac{2}{4}} \cdot 3^{\frac{3}{3}} \quad && \require{color}\color{red}{\text{(+1p)}}\\ =&\ 3^{\frac{1}{2}+\frac{1}{2}+1}\\ =& \ 3^2 \\ =&\underline{\underline{\ 9\ }} \quad &&\require{color}\color{red}{\text{(+1p)}} \end{align*}\)
\(\begin{align*}\textbf{b) } x^{\frac{1}{9}} \cdot x^{\frac{2}{9}}&=3 \\ x^{\frac{1}{9}+\frac{2}{9}}&=3 \\ x^{\frac{3}{9}}&=3 && \require{color}\color{red}{\text{(+1p)}}\\ x^{\frac{1}{3}}&=3 && ||( \ \ )^3\\ \Big(x^{\frac{1}{3}}\Big)^3 &= 3^3 && \require{color}\color{red}{\text{(+1p)}}\\ x^{\frac{1}{3} \cdot 3} &=27\\ x&=\underline{\underline{27}} && \require{color}\color{red}{\text{(+1p)}} \end{align*} \)
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