1. Explain how you know that \(\sqrt{37}\) is a little more than 6.

2. Explain how you know that \(\sqrt{95}\) is a little less than 10.

3. Explain how you know that \(\sqrt{30}\) is between 5 and 6.

The equation \(x^2=25\) has two solutions. This is because both \(5 \boldcdot 5 = 25\), and also \(\text-5 \boldcdot \text-5 = 25\). So 5 is a solution, and \(\text-5\) is also a solution.

Select all the equations that have a solution of \(\text-4\):

Select all the irrational numbers in the list.

• \(\frac23\)
• \(\frac {\text{-}123}{45}\)
• \(\sqrt{14}\)
• \(\sqrt{64}\)
• \(\sqrt\frac91\)
• \(\text-\sqrt{99}\)
• \(\text-\sqrt{100}\)

Each grid square represents 1 square unit. What is the exact side length of the shaded square?

A square, not aligned to the horizontal or vertical gridlines, is on a square grid. The square is drawn such that the first vertex of the square is on the left side. The second vertex is 2 grid squares up and 3 grid squares right from the first vertex. The third vertex is 3 grid squares down and 2 grid squares right from the second vertex. The fourth vertex is 2 grid squares down and 3 grid squares left from the third vertex. The first vertex is 3 grid squares up and 2 grid squares left from the fourth vertex.

For each pair of numbers, which of the two numbers is larger? Estimate how many times larger.

1. \(0.37 \boldcdot 10^6\) and \(700 \boldcdot 10^4\)
2. \(4.87 \boldcdot 10^4\) and \(15 \boldcdot 10^5\)
3. \(500,000\) and \(2.3 \boldcdot 10^8\)