Solution 3.1:2e

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Looking first at
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Looking first at <math>\sqrt{18}</math> this square root expression can be simplified by writing 18 as a product of its smallest possible integer factors
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<math>\sqrt{18}</math>
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this square root expression can be simplified by writing
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<math>\text{18}</math>
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as a product of its smallest possible integer factors
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<math>18=2\centerdot 9=2\centerdot 3\centerdot 3=2\centerdot 3^{2}</math>
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{{Displayed math||<math>18 = 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}</math>}}
and then we can take the quadratic out of the square root sign by using the rule
and then we can take the quadratic out of the square root sign by using the rule
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<math>\sqrt{a^{2}b}=a\sqrt{b}</math>,
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<math>\sqrt{a^{2}b}=a\sqrt{b}</math> (valid for non-negative ''a'' and ''b''),
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{{Displayed math||<math>\sqrt{18} = \sqrt{2\cdot 3^{2}} = 3\sqrt{2}\,\textrm{.}</math>}}
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<math>\sqrt{18}=\sqrt{2\centerdot 3^{2}}=3\sqrt{2}</math>
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In the same way, we write <math>8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3}</math> and get
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In the same way, we write
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<math>8=2\centerdot 4=2\centerdot 2\centerdot 2=2^{3}</math>
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and get
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<math>\sqrt{8}=\sqrt{2\centerdot 2^{2}}=2\sqrt{2}</math>
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{{Displayed math||<math>\sqrt{8} = \sqrt{2\cdot 2^{2}} = 2\sqrt{2}\,\textrm{.}</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{18}\sqrt{8}
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& \sqrt{18}\sqrt{8}=3\sqrt{2}\centerdot 2\sqrt{2}=3\centerdot 2\centerdot \left( \sqrt{2} \right)^{2} \\
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&= 3\sqrt{2}\cdot 2\sqrt{2}\\[5pt]
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& =3\centerdot 2\centerdot 2=12 \\
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&= 3\cdot 2\cdot \bigl(\sqrt{2}\bigr)^{2}\\[5pt]
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& \\
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&= 3\cdot 2\cdot 2\\[5pt]
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\end{align}</math>
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&= 12\,\textrm{.}
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\end{align}</math>}}

Current revision

Looking first at \displaystyle \sqrt{18} this square root expression can be simplified by writing 18 as a product of its smallest possible integer factors

\displaystyle 18 = 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}

and then we can take the quadratic out of the square root sign by using the rule \displaystyle \sqrt{a^{2}b}=a\sqrt{b} (valid for non-negative a and b),

\displaystyle \sqrt{18} = \sqrt{2\cdot 3^{2}} = 3\sqrt{2}\,\textrm{.}

In the same way, we write \displaystyle 8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3} and get

\displaystyle \sqrt{8} = \sqrt{2\cdot 2^{2}} = 2\sqrt{2}\,\textrm{.}

All together, we get

\displaystyle \begin{align}

\sqrt{18}\sqrt{8} &= 3\sqrt{2}\cdot 2\sqrt{2}\\[5pt] &= 3\cdot 2\cdot \bigl(\sqrt{2}\bigr)^{2}\\[5pt] &= 3\cdot 2\cdot 2\\[5pt] &= 12\,\textrm{.} \end{align}

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