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

1. Complete the table with the area of each square in square units, and its exact side length in units.
 figure area side length A B C D E
2. This table includes areas in square units and side lengths in units of some more squares. Complete the table.
 area side length 9 23 89 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

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 $$\sqrt{b}$$

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 $$\sqrt[3]{a}$$: