Lesson 2
Square Roots and Cube Roots
 Let’s think about square and cube roots.
2.1: It’s a Square
Find the area of square \(ABCD\).
2.2: Squares and Their Side Lengths
 Complete the table with the area of each square in square units, and its exact side length in units.
figure A B C D E area side length  This table includes areas in square units and side lengths in units of some more squares. Complete the table.
area 9 23 89 side length 4 6.4
In the first question, all of the squares have vertices at grid points.
 Is there a square whose vertices are at grid points and whose area is 7 square units? Explain how you know.
 Is there a square whose vertices are at grid points and whose area is 10 square units? Explain how you know.
2.3: Cube It
 A cube has edge length 3 units. What is the volume of the cube?
 A cube has edge length 4 units. What is the volume of the cube?
 A cube has volume 8 units. What is the edge length of the cube?
 A cube has volume 7 units. What is the edge length of the cube?

\(\sqrt[3]{1,\!200}\) is between 10 and 11 because \(10^3 = 1,\!000\) and \(11^3 = 1,\!331\). Determine the whole numbers that each of these cube roots lies between:
 \(\sqrt[3]{5}\)
 \(\sqrt[3]{10}\)
 \(\sqrt[3]{50}\)
 \(\sqrt[3]{100}\)
 \(\sqrt[3]{500}\)
between  1 and 2  2 and 3  3 and 4  4 and 5  5 and 6  6 and 7  7 and 8  8 and 9 

Summary
If a square has side length \(s\), then the area is \(s^2\). If a square has area \(A\), then the side length is \(\sqrt{A}\). For a positive number \(b\), the square root of \(b\) is defined as the positive number that squares to make \(b\), and it is written as \(\sqrt{b}\). In other words, \(\left(\sqrt{b}\right)^2 = b\). We can also think of \(\sqrt{b}\) as a solution to the equation \(x^2 = b\). This square has an area of \(b\) because its sides have length \(\sqrt{b}\):
Similarly, if a cube has edge length \(s\), then the volume is \(s^3\). If a cube has volume \(V\), then the edge length is \(\sqrt[3]{V}\). The number \(\sqrt[3]{a}\) is defined as the number that cubes to make \(a\). In other words, \(\left(\sqrt[3]{a}\right)^3 = a\). We can also think of \(\sqrt[3]{a}\) as a solution to the equation \(x^3 = a\). This cube has a volume of \(a\) because its sides have length \(\sqrt[3]{a}\):