Question

Essential Question How do you evaluate
cube roots and square roots?
2. Generalize A certain number is both a perfect
square and a perfect cube. Will its square root
and its cube root always be different numbers?
Explain.
3. Critique Reasoning A cube-shaped box has
a volume of 27 cubic inches. Bethany says each
side of the cube measures 9 inches because
9 × 3 = 27. Is Bethany correct? Explain your
reasoning.
Do You Know How?
4. A cube has a volume of 8 cubic inches. What is
the length of each edge of the cube?
5. Below is a model of the infield of a baseball
stadium. How long is each side of the infield?
A = 81 in.²
6. Julio cubes a number and then takes the cube
root of the result. He ends up with 20. What
number did Julio start with

Answer 1

Essential idea — how to evaluate square roots and cube roots
- A square root undo a squaring: if s^2 = A, then s = √A (the principal square root √A is nonnegative). Example: √36 = 6 because 6^2 = 36.
- A cube root undoes cubing: if c^3 = A, then c = ∛A. Cube roots may be negative: ∛27 = 3 and ∛(−8) = −2 because (−2)^3 = −8.

2. Generalize
- If a number N is both a perfect square and a perfect cube, then N = k^6 for some integer k (lcm(2,3)=6).
- Its square root is k^3 and its cube root is k^2. These are equal only when k^3 = k^2 ⇔ k = 0 or k = 1.
- So the square root and cube root are different for every k > 1; they are the same only for N = 0 and N = 1.

3. Critique reasoning
- Bethany is incorrect. Volume of a cube = side^3. If volume = 27, side = ∛27 = 3 inches (because 3×3×3 = 27). Her 9×3 = 27 uses two factors, not three equal side lengths.

4.
- Volume = 8, so side = ∛8 = 2 inches.

5.
- If the infield is a square with area A = 81 in^2, each side = √81 = 9 inches.

6.
- Cubing a number x then taking its cube root returns x. If Julio ends up with 20, he started with 20.