Wortels basis

Leerdoelen

•Je kunt uitleggen wat een wortel is.

•Je kunt de rekenregels van wortels uitleggen.

•Je kunt rekenen met wortels.

Wat is een wortel?

Van welk positief getal is het kwadraat gelijk aan 16? Als je bedenkt dat we een getal moeten vinden dat, vermenigvuldigd met zichzelf, 16 zou maken, dan zou dat getal 4 moeten zijn. We zeggen simpelweg dat de wortel van 16 is 4.

Enkele wortelregels

Er zijn een paar regels die we moeten volgen bij het werken met wortels. Ten eerste, de wortel van een getal kan nooit negatief zijn. Ook al is -4 vermenigvuldigd met -4 gelijk aan 16, de wortel van 16 is niet -4!

Tabel met wortels

Het is nuttig om een tabel te hebben met de wortels van verschillende getallen. Probeer deze tabel uit je hoofd te leren:

Wortels bij optellen en aftrekken

Wat is het verschil tussen \sqrt{9} + \sqrt{16} en \sqrt{9 + 16} ?

Als we eerst de wortel van 16 en die van 9 afzonderlijk uitrekenen en dan optellen, krijgen we 7. Maar als we eerst 9 en 16 optellen en dan de wortel nemen, krijgen we 5.

Dit laat zien dat de wortel van een som niet gelijk is aan de som van de wortels.

Bij het optellen en aftrekken van wortels geldt dus:

• \sqrt{a} + \sqrt{b} \neq \sqrt{a + b}

• \sqrt{a} - \sqrt{b} \neq \sqrt{a - b}

Wortels bij vermenigvuldigen en delen

Interessant genoeg, wanneer we met vermenigvuldiging werken, zien we dat het antwoord wel hetzelfde is, of de waarden samen of apart in de wortel staan.

Ook bij delen met wortels maakt het niet uit of we de waarden al dan niet samen in de wortel houden.

Bij het vermenigvuldigen en delen van wortels geldt dus:

• \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}

•\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\large{\frac{\sqrt{a}}{\sqrt{b}}}\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\large={\frac{\sqrt{a}}{\sqrt{b}}}\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}}\large{\sqrt{\frac{a}{b}}}={\frac{\sqrt{a}}{\sqrt{b}}} \large{\sqrt{\frac{a}{b}}} = \large{\frac{\sqrt{a}}{\sqrt{b}}}

Rekenen met kwadraten en wortels

Met wortels en kwadraten is er een specifieke volgorde die gevolgd moet worden bij het oplossen van bewerkingen:

1.Haakjes.

2.Kwadraten en wortels (let erop dat kwadrateren en de wortel van een getal trekken elkaars tegenovergestelde zijn, deze heffen elkaar dus op bij hetzelfde getal).

3.Vermenigvuldigen en delen (van links naar rechts).

4.Optellen en aftrekken (van links naar rechts).

Voorbeeld

Opgave: 12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7\frac12=\frac1212\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7\frac12=\frac{1}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7\frac12=112\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7\frac12=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7\frac1212\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-\frac1212\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-\frac{1}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-\frac{17}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-\frac{7}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-712\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7{,}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7{,}512\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-7{,}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-712\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=8-12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=812\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{10}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{1}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{75}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{7}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-\frac{\placeholder{}}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=8-12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=812\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=812\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{10}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{1}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot15}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot1}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5\cdot}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{5}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-\frac{\placeholder{}}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=8-12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=812\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{10}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{1}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{15}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{1}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot\frac{\placeholder{}}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-5\cdot12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-512\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}-12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{3}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=\frac{24}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=2412\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=212\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{10}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{1}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot\frac{15}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot1512\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot112\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-5\cdot12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-512\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}-12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{3}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot2}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12\cdot}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{12}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{1}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=\frac{\placeholder{}}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=112\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=1212\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=112\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{10}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{1}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{15}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot1512\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot112\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-512\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5x12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-512\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac2312\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac{2}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot212\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=1212\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=112\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}=12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{100}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{10}}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt1}12\cdot\frac23-5\cdot\frac{\sqrt{225}}{\sqrt{\placeholder{}}}12\cdot\frac23-5\cdot\frac{\sqrt{22}}{\sqrt{\placeholder{}}}12\cdot\frac23-5\cdot\frac{\sqrt2}{\sqrt{\placeholder{}}}12\cdot\frac23-5\cdot\frac{\sqrt{\placeholder{}}}{\sqrt{\placeholder{}}}12\cdot\frac23-5\cdot\frac{\sqrt{\placeholder{}}}{\placeholder{}}12\cdot\frac23-5\cdot\frac{\placeholder{}}{\placeholder{}}12\cdot\frac23-5\cdot12\cdot\frac23-512\cdot\frac23-12\cdot\frac2312\cdot\frac{2}{\placeholder{}}12\cdot212\cdot121