Answers and Solutions

1. 1. 1. \(\sqrt{49}=7\), since \(7^2 = 49\)
      2. \(\sqrt{81}=9\), since \(9^2=81\)
      3. \(\sqrt{36}=6\), since \(6^2=36\)
      4. \(\sqrt{121}=11\), since \(11^2=121\)
   2. Answers will match part a).
2. The value \(\sqrt{25}\) has an exact value because \(25\) is a perfect square that can be written as \(5 \times 5\) or \(5^2\).  The integer \(40\) is not a perfect square because there is not an exact value for \(a\) such that \(a^2=40\). As a result, we can only estimate the square root, and find an approximate value.
3. 1. 1. \(\sqrt{108}=10.4\)
      2. \(\sqrt{128}=11.3\)
      3. \(\sqrt{77}=8.8\)
   2. Answers will match part a).
4. 
    
5. To find the side length of the frame, we must find a value for \(l\) where \(l \times l = l^2 = 96\).
   \(\begin{align*} l &= \sqrt{96} \\ &= 9.8 \end{align*}\)
   Therefore, the picture frame has a side length of \(9.8\) cm.
6. 1. \(7.071 \times 7.071 = 7.071^2 = 49.999041\). Since Tarek took the square root of a positive integer, he likely took the square root of \(50\).
   2. \(7.071\) is an approximation because \(50\) is not a perfect square, which means that \(\sqrt{50}\) does not have an exact value.
7. 1. \(36 = 2^2 \times 3^2 = (2\times 3) \times (2 \times 3)\), so \(\sqrt{36}=2 \times 3\)
   2. \(256 = 2^8 = (2\times2\times2\times2) \times (2\times2\times2\times2) = 2^4 \times 2^4 = \), so \(\sqrt{256} = 2\times2\times2\times2 = 2^4\)
   3. \(196 = 2^2 \times 7^2 = (2 \times 7) \times (2 \times 7)\), so \(\sqrt{196}=2 \times 7\)
   4. \(324 =2^2 \times 3^4 =(2\times3\times3) \times (2\times3\times3)= (2 \times 3^2) \times (2\times 3^2)\), so \(\sqrt{324} =2\times3\times3= 2 \times 3^2\)
8. 1. If a number is divisible by \(6\), then it must be divisible by both \(2\) and \(3\).
      • Since the ones digit is \(6\), we know that the number \(7 ~\boxed{\phantom\square}~5~6\) will be divisible by \(2\) regardless of the digit that replaces \(\boxed{\phantom\square}\).
      • However, for the number \(7 ~\boxed{\phantom\square}~5~6\) to be divisible by \(3\), the sum of the digits must be divisible by \(3\). This means that \( 7 + \boxed{\phantom\square} + 5 + 6 = 18 + \boxed{\phantom\square}\) must be divisible by \(3\).  This is true only when \(\boxed{\phantom\square} = 0,~3,~6, \text{ or } 9\).
   2. 1. Recall that \(80^{2} = 6400\) and \(90^{2} = 8100\). Since the perfect square \(7 ~\boxed{\phantom\square}~5~6\) is between \(6400\) and \(8100\), it must be the square of an integer that is between \(80\) and \(90\). Also, this integer must be \(84\) or \(86\) because these are the only two integers in this range that result in a ones digit of \(6\) when multiplied by itself. Checking, we see that \(84^2 = 7056\) and \(86^2 = 7396\). Therefore, the value of \(\boxed{\phantom\square}\) is \(0\).
      2. Since \(84^2 = 7056\) it follows that \(\sqrt{7056} = 84\). 
9. Consider the following sequence of numbers.\[5^2,~ 15^2, ~25^2, ~35^2,~ 45^2, ~55^2,~65^2, ~75^2,~ 85^2,~ 95^2, \ldots\]
   1. \(25,~ 225, ~625, ~1225, ~2025\)
   2. \(3025, ~4225, ~5625,~ 7225, ~9025\)
   3. Notice that when each number is broken down, a pattern emerges.  The last two digits are always \(25\), and the leading digits increase by \(2\), \(4\), \(6\), and so on.  The diagram illustrates this observation.
      
      Finding the general term of this sequence is significantly more challenging.  
      We start by writing each number in the sequence using their prime factorization.

      Term Number	Term Value	Prime Factorization	Relationship
      \(1\)	\(5^2 =25\)	\(5^2\)	\(1 \times 5^2\)
      \(2\)	\(15^2 =225\)	\(3^2\times 5^2\)	\(3^2 \times 5^2\)
      \(3\)	\(25^2 =625\)	\(5^4\)	\(5^2\times 5^2\)
      \(4\)	\(35^2=1225\)	\(5^2\times7^2\)	\(7^2\times 5^2\)
      \(5\)	\(45^2=2025\)	\(3^4\times 5^2\)	\(9^2\times 5^2\)

      Notice that values we multiply \(25\) by in each case are the increasing sequence of odd perfect squares. Through some arithmetic, we can show the expression \(25(2n-1)^2\) represents the value of term \(n\) in the sequence.

   4. We can continue the sequence to test whether the result holds for three-digit numbers ending in \(5\).

      Term Number	Term Value	Relationship
      \(10\)	\(9025\)	\(361 \times 25 = 19^2 \times 25\)
      \(11\)	\(11~025\)	\(441 \times 25 = 21^2 \times 25\)
      \(12\)	\(13~225\)	\(529 \times 25 = 23^2 \times 25\)
      \(13\)	\(15~625\)	\(625 \times 25 = 25^2 \times 25\)

      It looks like the pattern continues and the rule we defined should continue to work. As an additional exercise, look at the last three digits of the \(11^{th}\), \(12^{th}\), and \(13^{th}\) term in the table. What do you notice?