Leerdoelen
•Je weet wat een macht met een gebroken exponent is.
•Je kunt een macht met een gebroken exponent herleiden tot een (hogeremachts)wortel.
•Je kunt de bijbehorende rekenregels toepassen.
Rekenregels voor machten met gebroken exponenten
•Macht van een macht: Als je een macht hebt en die moet je weer tot een andere macht verheffen, dan vermenigvuldig je de exponenten. Bijvoorbeeld,\left(a^{p}\right)^{q}=a^{p\cdot q}.\left(a^{p}\right)=a^{p\cdot q}.\left(a^{p}=a^{p\cdot q}.\right)\left(a^{p^{}}=a^{p\cdot q}.\right)\left(a^{p^{q}}=a^{p\cdot q}.\right)\left(a^{p)^{q}}=a^{p\cdot q}.\right)\left(a^{p^{q}}=a^{p\cdot q}.\right)a^{p^{q}}=a^{p\cdot q}.(a^{p^{q}}=a^{p\cdot q}.
•Wortel als macht: De q-de machtswortel van een getal kan worden geschreven als dat getal tot de macht \frac{1}{q}.(\frac{1}{q}.Dusis hetzelfde als de q-de machtswortel van,\sqrt[q]{x}. Als het een breuk is met ook een getal in de teller:
Voorbeelden van herleiden
Voorbeeld 1: Herleid\left(x^{\frac15}\right)^5.\left(x^{\frac15}\right)^5\left(x^{\frac15}\right)x^{\frac15}\left(x^{\frac15}\right)x^{\frac15}\left(x^{\frac{1}{\placeholder{}}}\right)x^{\frac15}\left(x^1\right)x^{\frac15}\left(\frac{x^1}{}\right)x^{\frac15}\left(\frac{x^1}{5}\right)x^{\frac15}\left(\frac{x^1}{\placeholder{}}\right)x^{\frac15}\left(x^1\right)x^{\frac15}\left(x\right)x^{\frac15}\left(\right)\left(x^{\frac15}\right.\left(\right)\left(x^{\frac15}\right)\left(\right)\left(x^{\frac15}\right))\left(x^{\frac15}\right)\left(x^{\frac15}\right)\left(\left(x^{\frac15}\right)\right)\left(f\left(x^{\frac15}\right)\right)\left(\left(x^{\frac15}\right)\right)\left(x^{\frac15}\right)x^{\frac15}\left(x^{\frac15}\right)\left(\left(x^{\frac15}\right)\right)\left(x^{\frac15}\right)x^{\frac15}\left(x^{\frac15}\right.\left(x^{\frac15}\right)\left(x^{\frac15}\right)^{}\left(x^{\frac15}\right)^5\left(x^{\frac15}\right)^5.\left(x^{\frac15}\right)^{}.\left(x^{\frac15}\right)^2.\left(x^{\frac15}\right).x^{\frac15}).x^{\frac15}.x^{\frac{1}{5)}}.x^{\frac{1}{5}}.x^{\frac{1}{5}5}.(x^{\frac{1}{5}5}.
Gebruik de regel van macht van een macht:\left(x^{\frac{1}{5}}\right)^5=x^{\frac{1}{5}\cdot5}=x^1=x.\left(x^{\frac{1}{5}}\right)^5=x^{\frac{1}{5}\cdot5}=x^1=x).
Voorbeeld 2: Schrijf de vijfde machtswortel vanals een macht.
De vijfde machtswortel en tot de macht vijf zijn tegengestelde bewerkingen, dus\sqrt[5]{x^5}=x.(\sqrt[5]{x^5}=x.
Breuken in exponenten
Wat als de exponent geen 1 is, maar een ander getal? Bijvoorbeeld,x^{\frac{p}{q}}.(x^{\frac{p}{q}}.
Dit kan worden herschreven alsx^{p\cdot\frac{1}{q}},(x^{p\cdot\frac{1}{q}},wat weer kan worden geschreven als\left(x^{p}\right)^{\frac{1}{q}}.(\left(x^{p}\right)^{\frac{1}{q}}.
Dit betekent datgelijk is aan de q-de machtswortel van,\sqrt[q]{x^{p}}\sqrt[q]{x}\sqrt[q]{}\sqrt[q]{p}.
Voorbeeldopdrachten
Opdracht 1: Schrijfals een macht vanx.(x.
De wortel vanx,x\sqrt{x},\sqrt{x}isx^{\frac{1}{2}}.(x^{\frac{1}{2}}.
Vermenigvuldig de machten:x^3\cdot x^{\frac{1}{2}}=x^{3+\frac{1}{2}}=x^{3.5}.(x^3\cdot x^{\frac{1}{2}}=x^{3+\frac{1}{2}}=x^{3.5}.
Opdracht 2: Schrijf\sqrt[7]{x^4}\cdot x^5\sqrt[7]{x^4}x^5\sqrt[7]{x^4}*x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5x^5\cdot x^5\cdot x^5\cdot x^5\\cdot x^5\cdot x^5\cdot x^5\cdot x^5x\cdot x^5x^{}\cdot x^5als een macht vanx.(x.
De zevende machtswortel vanisx^{\frac{4}{7}}.x^{\frac{4}{7}}.x^{\frac{4}{7}}.(x^{\frac{4}{7}}.
Vermenigvuldig de machten:x^{\frac{4}{7}}\cdot x^5=x^{\frac{4}{7}+5}=x^{\frac{39}{7}}.x^{\frac{4}{7}}\cdot x^5=x^{\frac{4}{7}+5}=x^{\frac{4}{7}}\cdot x^5=x^{\frac{4}{7}+5}x^{\frac{4}{7}}\cdot x^5=x^{\frac{4}{7}+5}+x^{\frac{4}{7}}\cdot x^5=x^{\frac{4}{7}+5}x^{\frac{4}{7}}\cdot x^5=x^{\frac{4}{7}+5}.x^{\frac{4}{7}}\cdot x^5=x^{\frac{4}{7}+5}).
Negatieve en gebroken exponenten
Soms moeten er negatieve en gebroken exponenten omgezet worden naar wortels:
Opdracht 3: Schrijf\frac{\sqrt[3]{x}}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{}{x^5}\frac{x}{x^5}\frac{x}{x^5}x\frac{x}{x^5}x^{}\frac{x}{x^5}x^5\frac{x}{x}x^5\frac{x}{\placeholder{}}x^5xx^5als een macht vanx.(x.
De derde machtswortel vanisx^{\frac{1}{3}}.(x^{\frac{1}{3}}.
Deel de machten:\frac{x^{\frac13}}{x^5}=x^{\frac{1}{3}-5}=x^{-4\frac23}.\frac{x^{\frac13}}{x^5}=x^{\frac{1}{3}-5}=x^{-\frac23}.\frac{x^{\frac13}}{x^5}=x^{\frac{1}{3}-5}=x^{-\frac{}{3}}.\frac{x^{\frac13}}{x^5}=x^{\frac{1}{3}-5}=x^{-\frac13}.\frac{x^{\frac13}}{x^5}=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.\frac{x^{\frac13}}{x}=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.\frac{x^{\frac13}}{\placeholder{}}=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.x^{\frac{1}{3}}=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.x^{\frac{1}{3}}\div=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.x^{\frac{1}{3}}\div x=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.x^{\frac{1}{3}}\div x^{}=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.x^{\frac{1}{3}}\div x^5=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.(x^{\frac{1}{3}}\div x^5=x^{\frac{1}{3}-5}=x^{-\frac{14}{3}}.
Opdracht 4: Schrijf\frac{\sqrt[5]{x^2}}{\sqrt[4]{x}}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{s}\frac{\sqrt[5]{x^2}}{sq}\frac{\sqrt[5]{x^2}}{sqr}\frac{\sqrt[5]{x^2}}{sqrt}\frac{\sqrt[5]{x^2}}{sqrt[}\frac{\sqrt[5]{x^2}}{sqrt[4}\frac{\sqrt[5]{x^2}}{sqrt[4]}\frac{\sqrt[5]{x^2}}{sqrt[4]{x}}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{}\frac{\sqrt[5]{x^2}}{\placeholder{}}\sqrt[5]{x^2}als een macht vanx.(x.
De vijfde machtswortel vanisx^{\frac{2}{5}}.(x^{\frac{2}{5}}.
De vierde machtswortel vanisx^{\frac{1}{4}}.(x^{\frac{1}{4}}.
Deel de machten:x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=x^{\frac{3}{20}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=x^{\frac32}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=x^{\frac{3}{}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=x^{\frac31}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=x^{\frac{3}{\placeholder{}}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=x^3.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=x.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}=.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{20}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac52}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{2-}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac52}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-\frac{5}{\placeholder{}}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-5}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}-}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{20}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac82}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^{\frac{8}{\placeholder{}}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^8.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=\frac{x^8}{}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=\frac{x^8}{2}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=\frac{x^8}{\placeholder{}}.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x^8.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=x.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}=.x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}.(x^{\frac{2}{5}}\div x^{\frac{1}{4}}=x^{\frac{2}{5}-\frac{1}{4}}.
Uitdrukkingen zonder negatieve en gebroken exponenten
Opdracht 5: Schrijf 5a^{3\frac12}5a^{3\left(\frac12\right.}5a^{3\left(\frac12\right)}5a^{3\left(\frac12\right)}5a^{3\left(\frac{1}{\placeholder{}}\right)}5a^{3\left(1\right)}5a^{3\left(\right)}5a^35a^{31}5a^{\frac{31}{}}5a^{\frac{31}{2}}5a^{\frac{31}{\placeholder{}}}5a^{31}5a^35a^35a^{}5a^15a^35a^315a^35a^{}5a^{^{}}5a^{^3}5a^{^31}5a^{^3}5a5zonder negatieve en zonder gebroken exponenten.
5a^{3\frac12}=5a^3\cdot a^{\frac12}=5a^3\sqrt{a}5a^{3\frac12}=5a^3\cdot a^{\frac12}=5a^3\sqrt{\placeholder{}}5a^{3\frac12}=5a^3\cdot a^{\frac12}=5a^35a^{3\frac12}=5a^3\cdot a^{\frac12}=5a5a^{3\frac12}=5a^3\cdot a^{\frac12}=55a^{3\frac12}=5a^3\cdot a^{\frac12}=5a^{3\frac12}=5a^3\cdot a^{\frac12}5a^{3\frac12}=5a^3\cdot a^{\frac{1}{\placeholder{}}}5a^{3\frac12}=5a^3\cdot a^15a^{3\frac12}=5a^3\cdot\frac{a^1}{\placeholder{}}5a^{3\frac12}=5a^3\cdot a^15a^{3\frac12}=5a^3\cdot a5a^{3\frac12}=5a^3\cdot5a^{3\frac12}=5a^35a^{3\frac12}=5a5a^{3\frac12}=55a^{3\frac12}=5a^{3\frac12}5a^{3\frac12}i5a^{3\frac12}io5a^{3\frac12}iok5a^{3\frac12}iokd5a^{3\frac12}iokd5a^{3\frac12}io5a^{3\frac12}io5a^{3\frac12}5a^{3\frac12}5a^{3\left(\frac12\right.}5a^{3\left(\frac12\right)}5a^{3\left(\frac12\right)}5a^{3\left(\frac{1}{}\right)}5a^{3\left(\frac13\right)}5a^{3\left(\frac{1}{32}\right)}5a^{3\left(\frac13\right)}5a^{3\left(\frac{1}{\placeholder{}}\right)}5a^{3\left(1\right)}5a^{3\left(\right)}5a^35a^315a^35a^3j5a^35a^{}5a^{\#}5a^{}5a^15a^35a^315a^35a5
Opdracht 6: Schrijf \frac12a^{\frac13}b^{-\frac34}\frac12a^{\frac13}b^{-\frac34}z\frac12a^{\frac13}b^{-\frac34}zo\frac12a^{\frac13}b^{-\frac34}z\frac12a^{\frac13}b^{-\frac34}\frac12a^{\frac13}b^{-\frac{3}{\placeholder{}}}\frac12a^{\frac13}b^{-3}\frac12a^{\frac13}b^{-}\frac12a^{\frac13}b\frac12a^{\frac13}\frac12a^{\frac13}s\frac12a^{\frac13}sz\frac12a^{\frac13}sz\frac12a^{\frac13}\frac12a^{\frac{1}{\placeholder{}}}\frac12a^1\frac12a\frac12\frac{1}{2a}\frac12\frac{1}{\placeholder{}}1zonder negatieve en zonder gebroken exponenten.
\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\frac{\sqrt[3]{a}}{2\sqrt[4]{b^3}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\frac{\sqrt[3]{a}}{2\sqrt[4]{b}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\frac{\sqrt[3]{a}}{2\sqrt[4]{\placeholder{}}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\frac{\sqrt[3]{a}}{2\sqrt[\placeholder{}]{\placeholder{}}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\frac{\sqrt[3]{a}}{2}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\frac{\sqrt[3]{a}}{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\sqrt[3]{a}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\sqrt[3]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\sqrt[\placeholder{}]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}=\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{b}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[4]{\placeholder{}}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\sqrt[\placeholder{}]{\placeholder{}}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac{1}{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot1\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\cdot\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{1}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac{\sqrt[3]{a}}{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{a}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[1]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[\placeholder{}]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{}a\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{1}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{1}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{1}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[3]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\sqrt[\placeholder{}]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\cdot\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac{1}{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=1\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}=\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac34}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^{\frac{3}{\placeholder{}}}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b^3}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{b}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac{1}{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot1\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\cdot\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{a}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[3]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\sqrt[\placeholder{}]{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=\frac12\cdot\frac12a^{\frac13}b^{-\frac34}=\frac12\frac12a^{\frac13}b^{-\frac34}=\frac{1}{\placeholder{}}\frac12a^{\frac13}b^{-\frac34}=1\frac12a^{\frac13}b^{-\frac34}=\frac12a^{\frac13}b^{-\frac34}
