# Lesson 10

Edge Lengths, Volumes, and Cube Roots

Let’s explore the relationship between volume and edge lengths of cubes.

### Problem 1

1. What is the volume of a cube with a side length of
1. 4 centimeters?
2. $$\sqrt[3]{11}$$ feet?
3. $$s$$ units?
2. What is the side length of a cube with a volume of
1. 1,000 cubic centimeters?
2. 23 cubic inches?
3. $$v$$ cubic units?

### Problem 2

Write an equivalent expression that doesn’t use a cube root symbol.

1. $$\sqrt[3]{1}$$
2. $$\sqrt[3]{216}$$
3. $$\sqrt[3]{8000}$$
4. $$\sqrt[3]{\frac{1}{64}}$$
5. $$\sqrt[3]{\frac{27}{125}}$$
6. $$\sqrt[3]{0.027}$$
7. $$\sqrt[3]{0.000125}$$

### Problem 3

Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.

1. $$t^3=216$$

2. $$a^2=15$$

3. $$m^3=8$$

4. $$c^3=343$$

5. $$f^3=181$$

### Problem 4

For each cube root, find the two whole numbers that it lies between.

1. $$\sqrt[3]{11}$$
2. $$\sqrt[3]{80}$$
3. $$\sqrt[3]{120}$$
4. $$\sqrt[3]{250}$$

### Problem 5

Order the following values from least to greatest:

$$\displaystyle \sqrt[3]{530},\;\sqrt{48},\;\pi,\;\sqrt{121},\;\sqrt[3]{27},\;\frac{19}{2}$$

### Problem 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 also -5 is a solution. But! The equation $$x^3=125$$ only has one solution, which is 5. This is because $$5 \boldcdot 5 \boldcdot 5 = 125$$, and there are no other numbers you can cube to make 125. (Think about why -5 is not a solution!)

Find all the solutions to each equation.

1. $$x^3=8$$
2. $$\sqrt[3]x=3$$
3. $$x^2=49$$
4. $$x^3=\frac{64}{125}$$