Exercises

1.  Recall

   A number is written in scientific notation if it is in the form \(a\times10^n\), where \(a\) is a real number and \(1\le a\lt10\) and \(n\) is a non-zero integer. 

   Examples of numbers written in scientific notation are \(1.23\times10^9\) and \(8.7\times10^{-12}\).

   We can use exponent rules to simplify values written in scientific notation. Once we have performed a given operation, we should ensure that our final answer is also written in scientific notation.  The \(a \) value should have the same number of decimal places as the values in the the question.  For example:

   \(\begin{align*} (5.62\times10^6)\times(3.1\times10^7)&=(5.62\times3.1)\times(10^6\times10^7)\\ &=17.422\times10^{13}\\ &=(1.7422\times10)\times10^{13}\\ &=1.7422\times(10\times10^{13})\\ &=1.7422\times10^{14}\\ &=1.74\times10^{14} \end{align*}\)

   1. The diameter of the sun is approximately \(1.391\times10^6\) kilometres. The diameter of Earth is approximately \(12~742\) kilometres. Approximately how many Earths could fit along the diameter of the sun? 
   2. One water molecule weighs approximately \(2.989\times10^{-26}\) kilograms. How many water molecules are there in Lake Ontario, which contains approximately \(1.64\times10^{15}\) kilograms of water?
   3. Diatoms are a group of microalgae found in oceans, waterways, and soil. They are a primary source of food in the sea. One type of diatom measures \(0.07\) millimetres in length. How many diatoms would be needed to stretch along \(1200\) kilometres of shoreline? 
   4. In 2018, Canada's population was approximately \(3.706\times10^7\). It is estimated that the average Canadian used \(329\) litres of water per day for laundry, bathing, flushing, and drinking. Estimate how much water Canadians used per day in the year 2018.
2. Determine the value of the positive integer \(n\).
   1. \(\dfrac{n^8}{n^6}=25\)
   2. \(\dfrac{n^6}{n^3}=\dfrac{1}{8}\)
   3. \(n^5\times n^2\div n^3=10~000\)
3. Simplify the rational expression.

   \(\Large\dfrac{\overbrace{n \times n \times n \times \cdots \times n}^{n \text{ times}}}{\underbrace{n +n +n + \cdots + n}_{n \text{ terms}}}\)

4. Given \((2^x)(2^y)=128\), 
   1. for positive integers \(x\) and \(y\), how many possible values are there for \(x\)?
   2. for integers \(x\) and \(y\), how many possible values are there for \(x\)? 
5. Simplify \(\dfrac{444^{15}}{222^{14}}\) using exponent laws, then evaluate.
6. Write in exponential form \(\sqrt[3]{\sqrt[4]{\sqrt{\sqrt[5]{5}}}}\).
7. Given integers \(x\), \(y\), and \(z\) where \(x^4y^5z^6\lt0\), determine if each of \(x\), \(y\), and \(z\) are positive or negative. Justify your answer.
8. Determine the values of \(A\), \(B\), and \(C\).

   \(\dfrac{8^8\times15^3}{6^4\times30^8}=\dfrac{2^A}{3^B\times5^C}\)

9. This is Problem 1b from the 2010 Euclid Contest.

   If \(2^53^{13}5^9x=2^73^{14}5^9\), what is the value of \(x\)?

10. Determine all possible integers \(x\) such that \(\big(2x^2+5x-3\big)^{x^2-10x+24}=1\).