Lesson 10
Edge Lengths, Volumes, and Cube Roots
Problem 1
 What is the volume of a cube with a side length of
 4 centimeters?
 \(\sqrt[3]{11}\) feet?
 \(s\) units?
 What is the side length of a cube with a volume of
 1,000 cubic centimeters?
 23 cubic inches?
 \(v\) cubic units?
Problem 2
Write an equivalent expression that doesn’t use a cube root symbol.
 \(\sqrt[3]{1}\)
 \(\sqrt[3]{216}\)
 \(\sqrt[3]{8000}\)
 \(\sqrt[3]{\frac{1}{64}}\)
 \(\sqrt[3]{\frac{27}{125}}\)
 \(\sqrt[3]{0.027}\)
 \(\sqrt[3]{0.000125}\)
Problem 3
Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.

\(t^3=216\)

\(a^2=15\)

\(m^3=8\)

\(c^3=343\)

\(f^3=181\)
Problem 4
For each cube root, find the two whole numbers that it lies between.
 \(\sqrt[3]{11}\)
 \(\sqrt[3]{80}\)
 \(\sqrt[3]{120}\)
 \(\sqrt[3]{250}\)
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 \(\text5 \boldcdot \text5 = 25\). 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.
 \(x^3=8\)
 \(\sqrt[3]x=3\)
 \(x^2=49\)
 \(x^3=\frac{64}{125}\)