Solution 3.1:4c

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Each term in the expression can be simplified by breaking down the number under the root sign into its factors,
Each term in the expression can be simplified by breaking down the number under the root sign into its factors,
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{{Displayed math||<math>\begin{align}
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50 &= 5\cdot 10 = 5\cdot 5\cdot 2 = 2\cdot 5^{2}\,,\\[5pt]
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20 &= 2\cdot 10 = 2\cdot 2\cdot 5 = 2^{2}\cdot 5\,,\\[5pt]
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18 &= 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}\,,\\[5pt]
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80 &= 8\cdot 10 = (2\cdot 4)\cdot (2\cdot 5) = (2\cdot 2\cdot 2)\cdot (2\cdot 5) = 2^{4}\cdot 5\,,
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\end{align}</math>}}
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<math>\begin{align}
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and then taking the squares out from under the root sign,
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& 50=5\centerdot 10=5\centerdot 5\centerdot 2=2\centerdot 5^{2} \\
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& 20=2\centerdot 10=2\centerdot 2\centerdot 5=2^{2}\centerdot 5 \\
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& 18=2\centerdot 9=2\centerdot 3\centerdot 3=2\centerdot 3^{2} \\
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& 80=8\centerdot 10=\left( 2\centerdot 4 \right)\centerdot \left( 2\centerdot 5 \right)=\left( 2\centerdot 2\centerdot 2 \right)\centerdot \left( 2\centerdot 5 \right)=2^{4}\centerdot 5 \\
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\end{align}</math>
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and then taking the squares out from under the root sign.
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<math>\begin{align}
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& \sqrt{50}=\sqrt{2\centerdot 5^{2}}=5\sqrt{2} \\
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& \sqrt{20}=\sqrt{2^{2}\centerdot 5}=2\sqrt{5} \\
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& \sqrt{18}=\sqrt{2\centerdot 3^{2}}=3\sqrt{2} \\
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& \sqrt{80}=\sqrt{2^{4}\centerdot 5}=2^{2}\sqrt{5}=4\sqrt{5} \\
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\end{align}</math>
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{{Displayed math||<math>\begin{align}
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\sqrt{50} &= \sqrt{2\cdot 5^2} = 5\sqrt{2}\,,\\
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\sqrt{20} &= \sqrt{2^2\cdot 5} = 2\sqrt{5}\,,\\
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\sqrt{18} &= \sqrt{2\cdot 3^2} = 3\sqrt{2}\,,\\
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\sqrt{80} &= \sqrt{2^4\cdot 5} = 2^{2}\sqrt{5} = 4\sqrt{5}\,\textrm{.}
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\end{align}</math>}}
All together, we get
All together, we get
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{{Displayed math||<math>\begin{align}
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<math>\begin{align}
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\sqrt{50} + 4\sqrt{20} - 3\sqrt{18} - 2\sqrt{80}
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& \sqrt{50}+4\sqrt{20}-3\sqrt{18}-2\sqrt{80} \\
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&= 5\sqrt{2} + 4\cdot 2\sqrt{5} - 3\cdot 3\sqrt{2} - 2\cdot 4\sqrt{5}\\[5pt]
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& =5\sqrt{2}+4\centerdot 2\sqrt{5}-3\centerdot 3\sqrt{2}-2\centerdot 4\sqrt{5} \\
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&= 5\sqrt{2} + 8\sqrt{5} - 9\sqrt{2} - 8\sqrt{5}\\[5pt]
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& =5\sqrt{2}+8\sqrt{5}-9\sqrt{2}-8\sqrt{5} \\
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&= (5-9)\sqrt{2} + (8-8)\sqrt{5} = -4\sqrt{2}\,\textrm{.}
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& =\left( 5-9 \right)\sqrt{2}+\left( 8-8 \right)\sqrt{5}=-4\sqrt{2} \\
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\end{align}</math>}}
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\end{align}</math>
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Current revision

Each term in the expression can be simplified by breaking down the number under the root sign into its factors,

\displaystyle \begin{align}

50 &= 5\cdot 10 = 5\cdot 5\cdot 2 = 2\cdot 5^{2}\,,\\[5pt] 20 &= 2\cdot 10 = 2\cdot 2\cdot 5 = 2^{2}\cdot 5\,,\\[5pt] 18 &= 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}\,,\\[5pt] 80 &= 8\cdot 10 = (2\cdot 4)\cdot (2\cdot 5) = (2\cdot 2\cdot 2)\cdot (2\cdot 5) = 2^{4}\cdot 5\,, \end{align}

and then taking the squares out from under the root sign,

\displaystyle \begin{align}

\sqrt{50} &= \sqrt{2\cdot 5^2} = 5\sqrt{2}\,,\\ \sqrt{20} &= \sqrt{2^2\cdot 5} = 2\sqrt{5}\,,\\ \sqrt{18} &= \sqrt{2\cdot 3^2} = 3\sqrt{2}\,,\\ \sqrt{80} &= \sqrt{2^4\cdot 5} = 2^{2}\sqrt{5} = 4\sqrt{5}\,\textrm{.} \end{align}

All together, we get

\displaystyle \begin{align}

\sqrt{50} + 4\sqrt{20} - 3\sqrt{18} - 2\sqrt{80} &= 5\sqrt{2} + 4\cdot 2\sqrt{5} - 3\cdot 3\sqrt{2} - 2\cdot 4\sqrt{5}\\[5pt] &= 5\sqrt{2} + 8\sqrt{5} - 9\sqrt{2} - 8\sqrt{5}\\[5pt] &= (5-9)\sqrt{2} + (8-8)\sqrt{5} = -4\sqrt{2}\,\textrm{.} \end{align}

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