## 13.1: True or False: Cubed

Decide if each statement is true or false.

$\left( \sqrt[3]{5} \right)^3=5$

$\left(\sqrt[3]{27}\right)^3 = 3$

$7 = \left(\sqrt[3]{7}\right)^3$

$\left(\sqrt[3]{10}\right)^3 = 1,\!000$

$\left(\sqrt[3]{64}\right) = 2^3$

Let’s compare cube roots.

Decide if each statement is true or false.

$\left( \sqrt[3]{5} \right)^3=5$

$\left(\sqrt[3]{27}\right)^3 = 3$

$7 = \left(\sqrt[3]{7}\right)^3$

$\left(\sqrt[3]{10}\right)^3 = 1,\!000$

$\left(\sqrt[3]{64}\right) = 2^3$

What two whole numbers does each cube root lie between? Be prepared to explain your reasoning.

- $\sqrt[3]{5}$
- $\sqrt[3]{23}$
- $\sqrt[3]{81}$
- $\sqrt[3]{999}$

The numbers $x$, $y$, and $z$ are positive, and:

$$x^3= 5$$

$$y^3= 27$$

$$z^3= 700$$

- Plot $x$, $y$, and $z$ on the number line. Be prepared to share your reasoning with the class.
- Plot $\text- \sqrt[3]{2}$ on the number line.

Diego knows that $8^2=64$ and that $4^3=64$. He says that this means the following are all true:

- $\sqrt{64}=8$
- $\sqrt[3]{64}=4$
- $\sqrt{\text -64}=\text-8$
- $\sqrt[3]{\text -64}=\text -4$

Is he correct? Explain how you know.

Remember that square roots of whole numbers are defined as side lengths of squares. For example, $\sqrt{17}$ is the side length of a square whose area is 17. We define cube roots similarly, but using cubes instead of squares. The number $\sqrt[3]{17}$, pronounced “the cube root of 17,” is the edge length of a cube which has a volume of 17.

We can approximate the values of cube roots by observing the whole numbers around it and remembering the relationship between cube roots and cubes. For example, $\sqrt[3]{20}$ is between 2 and 3 since $2^3=8$ and $3^3=27$, and 20 is between 8 and 27. Similarly, since 100 is between $4^3$ and $5^3$, we know $\sqrt[3]{100}$ is between 4 and 5. Many calculators have a cube root function which can be used to approximate the value of a cube root more precisely. Using our numbers from before, a calculator will show that $\sqrt[3]{20} \approx 2.7144$ and that $\sqrt[3]{100} \approx 4.6416$.

Also like square roots, most cube roots of whole numbers are irrational. The only time the cube root of a number is a whole number is when the original number is a perfect cube.