Exercises

1. 1. Calculate the square root without a calculator.
      1. \(\sqrt{49}\)
      2. \(\sqrt{81}\)
      3. \(\sqrt{36}\)
      4. \(\sqrt{121}\)
   2. Use the square root, \(\sqrt{\phantom\square}\), button on your calculator to check your answers in part a).
2. Explain why \(\sqrt{25}\) has an exact value while for \(\sqrt{40}\) you can only give an approximate answer.
3. 1. Approximate each square root to one decimal place.
      1. \(\sqrt{108}\)
      2. \(\sqrt{128}\)
      3. \(\sqrt{77}\)
   2. Check your answers from part a) using the square root, \(\sqrt{\phantom\square}\), button on your calculator.
4. Place the following numbers in their approximate place on the number line.
   \[\sqrt{42} \quad\quad \sqrt{30} \quad\quad \sqrt{60} \quad\quad \sqrt{105} \quad\quad \sqrt{90}\]
5. The area of a small, square picture frame is \(96\) cm\(^2\). What is the side length of the frame (to one decimal place)?
6. Tarek took the square root of a positive integer and wrote down \(7.071\) in his notes.
   1. Find the number that Tarek took the square root of by multiplying \(7.071\) by itself.
   2. Is \(7.071\) an approximation for the square root of the positive integer? How do you know?
7. We can find the square root of a perfect square by splitting the prime factors into two equal and identical groups.
   For example, \(144 = 2^4 \times 3^2 = (2^2 \times 3) \times (2^2\times 3)\) and so \(\sqrt{144}=2^2\times3\).
   Using prime factorizations, find the value of each of the following numbers. Leave your answer written as a product of prime factors.
   1. \(\sqrt{36}\)
   2. \(\sqrt{256}\)
   3. \(\sqrt{196}\)
   4. \(\sqrt{324}\)
8. It is given that \(6\) is a factor of the number \(7 ~\boxed{\phantom\square}~5~6\), where \(\boxed{\phantom\square}\) represents a missing digit.
   1. Find all possible values for the missing number \(\boxed{\phantom\square}\).
   2. Suppose the number \(7 ~\boxed{\phantom\square}~5~6\) is a perfect square. 
      1. Find the value of the missing number \(\boxed{\phantom\square}\).
      2. Find the square root of the number.
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. Calculate the first five terms in the sequence.
   2. What would you expect the next five squares to be? Calculate the next five squares to check your hypothesis.
   3. What rule could you use to square any two-digit numbers ending in \(5\)?
   4. Does your rule work for three-digit numbers ending in \(5\)?