Exercises

1. Evaluate
   1. \(2^5\)
   2. \(3^3\)
   3. \(7^2\)
   4. \(10^6\)
2. Write the given number in expanded form using powers of \(10\) in exponential notation.
   1. \(413\)
   2. \(2018\)
   3. \(140~056\)
3. Order the numbers in each list from least to greatest. Which list takes the longest time to order? Explain.
   1. \(2^4,~ 2^{11},~ 2^{10},~ 2^7\)
   2. \(2^4,~ 5^4, ~3^4,~ 7^4\)
   3. \(2^7,~ 3^4,~ 4^3,~ 5^3\)
4. The list \(3,~6, ~10\) has the property that the sum of any two neighbouring numbers is a perfect square.  We verify that \(3+6 = 3^{2}\) and \(6 + 10 = 4^{2}\).
   1. Arrange the following numbers so that the sum of any two neighbouring numbers is a perfect square.
      \(7,~ 11,~ 9, ~14,~ 16,~ 2, ~25\)
   2. For a special challenge, find a way to arrange all of the integers from \(1\) to \(17\) into a list with the property from part a).
5. Peggy wants to build a solid cube with a side length of \(5\) units.  Peggy has \(45\) unit cubes.  How many more unit cubes does she need?
6. \(153\) is an example of a 3-digit Narcissistic number.  A 3-digit Narcissistic number has the property that the sum of the cubes of the 3 digits in the number is equal to the number itself.  For \(153\), we verify that \[\begin{align*} 153 & = 1^3 + 5^3 +3^3\\ & = 1 + 125 + 27\\ & = 153 \end{align*}\]
   1. Which of the following numbers are also 3-digit Narcissistic numbers?
      1. \(124\)
      2. \(371\)
      3. \(370\)
      4. \(407\)
   2. What property would a 4-digit Narcissistic number have?
7. Each shape in the diagrams below is a square with an integer side length.  The number in the middle of a square represents the area of that square.  Determine the area of all of the other squares in the diagrams.
   Diagram 1
   

   Diagram 2
   
8. Lagrange's four-square theorem states that every positive integer can be written as the sum of four or fewer square numbers.  For instance, \(23 = 3^2 + 3^2 + 2^2 + 1^2\) and \(30 = 5^2 + 2^2 + 1^2\).  Write each of the following integers as the sum of four or fewer square numbers.
   1. \(15\)
   2. \(24\)
   3. \(33\)
   4. any 3-digit, positive integer of your choosing
9. The four members of Chenille's group had different strategies for simplifying the product \(8^5 \times 8^9\) to a single power.  Their strategies are shown below. Explain the strategy used in each case to produce the simplified power.  Only one of these strategies provides a correct simplification of the product.  Identify this strategy and explain why it is correct.
   1. \(8^5 \times 8^9 \longrightarrow 8^{45}\)
   2. \(8^5 \times 8^9 \longrightarrow 64^{14}\)
   3. \(8^5 \times 8^9 \longrightarrow 8^{14}\)
   4. \(8^5 \times 8^9 \longrightarrow 64^{45}\)
10. The first eight square numbers are as follows: \(1,~ 4, ~9, ~16,~ 25, ~36, ~49,~ 64.\) The positive difference between the first two squares is \(4-1 = 3\) and between the second two squares is \(9-4 = 5\).
    1. Find the differences between each pair of consecutive square numbers in the list.
    2. What pattern do you see in the list of differences between consecutive square numbers?
    3. Verify that the pattern from part b) continues for the next four differences. 
    4. Can you use the following diagrams to explain why the pattern from part b) will continue forever?