Lesson 10

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?

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. \(\sqrt[3]{0.000125}\)

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. \(f^3=181\)

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. \(\sqrt[3]{250}\)

Order the following values from least to greatest:

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. \(x^3=\frac{64}{125}\)