Logarithms : Exercise (3.1) Solutions

တိုင်းတာမှုများပါသော တွက်ချက်မှုများ ဆောင်ရွက်ရာတွင် ရလဒ်ကို တိကျမှုအနည်းဆုံး ဖြစ်သော အတိုင်းအတာပမာဏအတိုင်း ဖေါ်ပြပေးသင့်ပါသည်။

ပေါင်းခြင်း နှင့် နုတ်ခြင်းတွင် ရလဒ်ကို တိကျမှုအနည်းဆုံး ဖြစ်သော အတိုင်းအတာ၏ ဒသမ အရေအတွက်အတိုင်း ဖေါ်ပြပေးသင့်ပါသည်။

မြှောက်ခြင်းနှင့် စားခြင်းတွင် ရလဒ်ကို တိကျမှုအနည်းဆုံး ဖြစ်သော အတိုင်းအတာ၏ အရာရောက်ဂဏန်း အရေအတွက်အတိုင်း ဖေါ်ပြပေးသင့်ပါသည်။

The Rules for Significant Figures

$\begin{array}{|l|l|l|l|} \hline \text { Example } & \begin{array}{c} \text { Scientific } \\ \text { Notation } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Significant } \\ \text { Figures } \end{array} & {\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text { Remark }} \\ \hline 0.00682 & 6.82 \times 10^{-3} & \ \ \ \ \ \ \ \ 3 & \text { Leading zeros are not significant. } \\ \hline 100.07 & 1.0007 \times 10^{2} & \ \ \ \ \ \ \ \ 5 & \text { Embedded zeros are always significant. } \\ \hline 300 & 3 \times 10^{2} & \ \ \ \ \ \ \ \ 1 & \begin{array}{l} \text { Trailing zeros are significant only if the } \\ \text { decimal point is specified. If the decimal } \\ \text { point is not specified, trailing zeros are } \\ \text { generally not significant. } \end{array} \\ \hline 300 . & 3.00 \times 10^{2} & \ \ \ \ \ \ \ \ 3 & \text { All digits are significant. } \\ \hline 300.0 & 3.000 \times 10^{3} & \ \ \ \ \ \ \ \ 4 & \text { All digits are significant. } \\ \hline 0.0080500 & \begin{array}{l} 8.0500 \times10^{-3} \\ \end{array} & \ \ \ \ \ \ \ \ 5 & \begin{array}{l} \text { Leading zeros are not significant whereas } \\ \text { embedded zeros and trailing zeros after } \\ \text { the decimal point are significant. } \end{array} \\ \hline \end{array}$

1.           How many significant figures are there in each of the following numbers?

              $\begin{array}{lll} \text{(a)}\ \ 2.175 & \text{(b)}\ \ 0.2175& \text{(c)}\ \ 0.0075\\\\ \text{(d)}\ \ 89400 & \text{(e)}\ \ 0.00046 & \text{(f)}\ \ 12.0500 & \end{array}$

Show/Hide Solution

$\begin{array}{lll} \text{(a)}\ \ 2.175 & \Rightarrow & 4\ \text{significant figures}\\\\ \text{(b)}\ \ 0.2175 & \Rightarrow & 4\ \text{significant figures}\\\\ \text{(c)}\ \ 0.0075 & \Rightarrow & 2\ \text{significant figures}\\\\ \text{(d)}\ \ 89400 & \Rightarrow & 3\ \text{significant figures}\\\\ \text{(e)}\ \ 0.00046 & \Rightarrow & 2\ \text{significant figures}\\\\ \text{(f)}\ \ 12.0500 & \Rightarrow & 6\ \text{significant figures}\\\\ \end{array}$

2.           Write in scientific notation.

              $\begin{array}{lll} \text{(a)}\ \ 24.86 & \text{(b)}\ \ 2.486& \text{(c)}\ \ 0.2486\\\\ \text{(d)}\ \ 0.002486 & \text{(e)}\ \ 0.073 & \text{(f)}\ \ 0.0086\\\\ \text{(g)}\ \ 0.934 & \text{(h)}\ \ 7 & \text{(i)}\ \ 6.843250\\\\ \text{(j)}\ \ 0.00056857 & \text{(k)}\ \ 62500 & \text{(l)}\ \ 3001 \end{array}$

Show/Hide Solution

$\begin{array}{lll} \text{(a)}\ \ 24.86 & = & 2.486\times 10^1 \\\\ \text{(b)}\ \ 2.486 & = & 2.486\times 10^0 \\\\ \text{(c)}\ \ 0.2486 & = & 2.486\times 10^{-1} \\\\ \text{(d)}\ \ 0.002486 & = & 2.486\times 10^{-3} \\\\ \text{(e)}\ \ 0.073 & = & 7.3\times 10^{-2} \\\\ \text{(f)}\ \ 0.0086 & = & 8.6\times 10^{-3} \\\\ \text{(g)}\ \ 0.934 & = & 9.34\times 10^{-1} \\\\ \text{(h)}\ \ 7 & = & 7\times 10^0 \\\\ \text{(i)}\ \ 6.843250 & = & 6.843250\times 10^0 \\\\ \text{(j)}\ \ 0.00056857& = & 5.6857\times 10^{-4} \\\\ \text{(k)}\ \ 62500 & = & 6.25\times 10^{4} \\\\ \text{(l)}\ \ 3001 & = & 3.001\times 10^3 \end{array}$

3.           Write each number in ordinary decimal form.

              $\begin{array}{l} \text{(a)}\ \ 7.84 \times 10^{4}\\\\ \text{(b)}\ \ 7.89 \times 10^{-4}\\\\ \text{(c)}\ \ 2.25 \times 10^{5}\\\\ \text{(d)}\ \ 4.01 \times 10^{-3} \end{array}$

Show/Hide Solution

$\begin{array}{lll} \text{(a)}\ \ 7.84 \times 10^{4} &=& 78400\\\\ \text{(b)}\ \ 7.89 \times 10^{-4} &=& 0.000789\\\\ \text{(c)}\ \ 2.25 \times 10^{5} &=& 225000\\\\ \text{(d)}\ \ 4.01 \times 10^{-3} &=& 0.00401 \end{array}$

4.           Simplify and give the answers in scientific notation.

              $\begin{array}{l} \text{(a)}\ \ 2.3 \times 10^{2}+1.7 \times 10^{2}\\\\ \text{(b)}\ \ 4.6 \times 10^{-3}-2.5 \times 10^{-3}\\\\ \text{(c)}\ \ \left(4.5 \times 10^{6}\right) \times\left(1.5 \times 10^{-2}\right)\\\\ \text{(d)}\ \ \displaystyle \frac{7.6 \times 10^{5}}{1.9 \times 10^{-2}} \end{array}$

Show/Hide Solution

$\begin{array}{l} \text{(a)}\;\;\ 2.3\times {{10}^{2}}+1.7\times {{10}^{2}}\\ \ \ \ =(2.3+1.7)\times {{10}^{2}}\\ \ \ \ =4.0\times {{10}^{2}}\\\\ \text{(b)}\;\;\ 4.6\times {{10}^{{-3}}}-2.5\times {{10}^{{-3}}}\\ \ \ \ =(4.6-2.5)\times {{10}^{{-3}}}\\ \ \ \ =2.1\times {{10}^{{-3}}}\\\\ \text{(c)}\;\;\;\left( {4.5\times {{{10}}^{6}}} \right)\times \left( {1.5\times {{{10}}^{{-2}}}} \right)\\ \ \ \ =(4.5\times 1.5)\times {{10}^{{6-2}}}\\ \ \ \ =6.8\times {{10}^{4}}\\\\ \text{(d)}\;\;\;\displaystyle \frac{{7.6\times {{{10}}^{5}}}}{{1.9\times {{{10}}^{{-2}}}}}\\ \ \ \ =4.0\times {{10}^{{5+2}}}\\ \ \ \ =4.0\times {{10}^{7}} \end{array}$

5.           Compute using scientific notation.

              $\begin{array}{l} \text{(a)}\ \ \displaystyle \frac{2.5 \times 10^{2}}{0.25 \times 0.002}\\\\ \text{(b)}\ \ \displaystyle \frac{33,000,000 \times 0.4}{1.1 \times 30}\\\\ \text{(c)}\ \ \displaystyle \frac{50 \times 0.014 \times 0.30}{10500}\\\\ \text{(d)}\ \ \displaystyle \frac{7000 \times 80 \times 300}{400} \end{array}$

Show/Hide Solution

$\begin{array}{l}\text{(a)}\;\;\ \displaystyle \frac{{2.5\times {{{10}}^{2}}}}{{0.25\times 0.002}}\\\ \ \ =\displaystyle \frac{{2.5\times {{{10}}^{2}}}}{{2.5\times {{{10}}^{{-1}}}\times 2.0\times {{{10}}^{{-3}}}}}\\\ \ \ =\displaystyle \frac{1}{2}\times {{10}^{{2+1+3}}}\\\ \ \ =0.5\times {{10}^{6}}\\\ \ \ =5.0\times {{10}^{5}}\\\\\text{(b)}\;\;\ \displaystyle \frac{{33,000,000\times 0.4}}{{1.1\times 30}}\\\ \ \ =\displaystyle \frac{{3.3\times {{{10}}^{7}}\times 4.0\times {{{10}}^{{-1}}}}}{{1.1\times 3.0\times {{{10}}^{1}}}}\\\ \ \ =\displaystyle \frac{{3.3\times 4.0}}{{3.3}}\times {{10}^{{7-1-1}}}\\\ \ \ =4.0\times {{10}^{5}}\\\\\text{(c)}\;\;\;\displaystyle \frac{{50\times 0.014\times 0.30}}{{10500}}\\\ \ \ =\displaystyle \frac{{5.0\times {{{10}}^{1}}\times 1.4\times {{{10}}^{{-2}}}\times 3.0\times {{{10}}^{{-1}}}}}{{1.05\times {{{10}}^{4}}}}\\\ \ \ =\displaystyle \frac{{5.0\times 1.4\times 3.0}}{{1.05}}\times {{10}^{{1-2-1-4}}}\\\ \ \ =\displaystyle \frac{{2.1\times 10}}{{1.05}}\times {{10}^{{-6}}}\\\ \ \ =2\times {{10}^{{-5}}}\\\\\text{(d)}\;\;\;\displaystyle \frac{{7000\times 80\times 300}}{{400}}\\\ \ \ =\displaystyle \frac{{7\times {{{10}}^{3}}\times 8\times {{{10}}^{1}}\times 3\times {{{10}}^{2}}}}{{4\times {{{10}}^{2}}}}\\\ \ \ =\displaystyle \frac{{7\times 8\times 3}}{4}\times {{10}^{{3+2+1-2}}}\\\ \ \ =42\times {{10}^{4}}\\\ \ \ =4.2\times {{10}^{5}}\end{array}$

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