Solution 3.1:2e
From Förberedande kurs i matematik 1
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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 |
- | < | + | |
- | {{ | + | {{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 | ||
+ | <math>\sqrt{a^{2}b}=a\sqrt{b}</math> (valid for non-negative ''a'' and ''b''), | ||
+ | |||
+ | {{Displayed math||<math>\sqrt{18} = \sqrt{2\cdot 3^{2}} = 3\sqrt{2}\,\textrm{.}</math>}} | ||
+ | |||
+ | In the same way, we write <math>8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3}</math> and get | ||
+ | |||
+ | {{Displayed math||<math>\sqrt{8} = \sqrt{2\cdot 2^{2}} = 2\sqrt{2}\,\textrm{.}</math>}} | ||
+ | |||
+ | All together, we get | ||
+ | |||
+ | {{Displayed math||<math>\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}</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} |