Solution 3.1:4d
From Förberedande kurs i matematik 1
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- | {{ | + | We start by factorizing the numbers under the root sign, |
- | < | + | |
- | {{ | + | {{Displayed math||<math>\begin{align} |
+ | 48 &= 2\cdot 24 = 2\cdot 2\cdot 12 = 2\cdot 2\cdot 2\cdot 6 = 2\cdot 2\cdot 2\cdot 2\cdot 3 = 2^{4}\cdot 3\,,\\ | ||
+ | 12 &= 2\cdot 6 = 2\cdot 2\cdot 3 = 2^{2}\cdot 3\,,\\ | ||
+ | 3 &= 3\,,\\ | ||
+ | 75 &= 3\cdot 25 = 3\cdot 5\cdot 5 = 3\cdot 5^{2}\,\textrm{.} | ||
+ | \end{align}</math>}} | ||
+ | |||
+ | Now, we can take the squares out from under the root signs, | ||
+ | |||
+ | {{Displayed math||<math>\begin{align} | ||
+ | \sqrt{48} &= \sqrt{2^4\cdot 3} = 2^2\sqrt{3} = 4\sqrt{3}\,,\\[5pt] | ||
+ | \sqrt{12} &= \sqrt{2^2\cdot 3} = 2\sqrt{3},\\[5pt] | ||
+ | \sqrt{3} &= \sqrt{3}\,,\\[5pt] | ||
+ | \sqrt{75} &= \sqrt{3\cdot 5^{2}} = 5\sqrt{3}\,, | ||
+ | \end{align}</math>}} | ||
+ | |||
+ | and then simplify the whole expression | ||
+ | |||
+ | {{Displayed math||<math>\begin{align} | ||
+ | \sqrt{48} + \sqrt{12} + \sqrt{3} - \sqrt{75} | ||
+ | &= 4\sqrt{3} + 2\sqrt{3} + \sqrt{3} - 5\sqrt{3}\\[5pt] | ||
+ | &= (4+2+1-5)\sqrt{3}\\[5pt] | ||
+ | &= 2\sqrt{3}\,\textrm{.} | ||
+ | \end{align}</math>}} |
Current revision
We start by factorizing the numbers under the root sign,
\displaystyle \begin{align}
48 &= 2\cdot 24 = 2\cdot 2\cdot 12 = 2\cdot 2\cdot 2\cdot 6 = 2\cdot 2\cdot 2\cdot 2\cdot 3 = 2^{4}\cdot 3\,,\\ 12 &= 2\cdot 6 = 2\cdot 2\cdot 3 = 2^{2}\cdot 3\,,\\ 3 &= 3\,,\\ 75 &= 3\cdot 25 = 3\cdot 5\cdot 5 = 3\cdot 5^{2}\,\textrm{.} \end{align} |
Now, we can take the squares out from under the root signs,
\displaystyle \begin{align}
\sqrt{48} &= \sqrt{2^4\cdot 3} = 2^2\sqrt{3} = 4\sqrt{3}\,,\\[5pt] \sqrt{12} &= \sqrt{2^2\cdot 3} = 2\sqrt{3},\\[5pt] \sqrt{3} &= \sqrt{3}\,,\\[5pt] \sqrt{75} &= \sqrt{3\cdot 5^{2}} = 5\sqrt{3}\,, \end{align} |
and then simplify the whole expression
\displaystyle \begin{align}
\sqrt{48} + \sqrt{12} + \sqrt{3} - \sqrt{75} &= 4\sqrt{3} + 2\sqrt{3} + \sqrt{3} - 5\sqrt{3}\\[5pt] &= (4+2+1-5)\sqrt{3}\\[5pt] &= 2\sqrt{3}\,\textrm{.} \end{align} |