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 \(ABCD\).

Square with tilted square ABCD inside of it, creating 4 right triangles with legs each measuring 2 and 3 units.

 

2.2: Squares and Their Side Lengths

  1. Complete the table with the area of each square in square units, and its exact side length in units.
    Squares A, B, C, D, E. A Side length = 1. B Side length = square root of 2, C Side length = 2, D Side length = square root of 5, E Side length = square root of 8.
    figure A B C D E
    area                              
    side length          
  2. 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.

  1. Is there a square whose vertices are at grid points and whose area is 7 square units? Explain how you know.
  2. 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

Colorful cubes of different sizes.
  1. A cube has edge length 3 units. What is the volume of the cube?
  2. A cube has edge length 4 units. What is the volume of the cube?
  3. A cube has volume 8 units. What is the edge length of the cube?
  4. A cube has volume 7 units. What is the edge length of the cube?
  5. \(\sqrt[3]{1,\!200}\) 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 \(\sqrt{b}\): 

Square with side length = square root of 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 \(\sqrt[3]{a}\):

Cube with side length = cube root of a