Question

^3√81 ^3√-64Simplify each cube root expression. Describe the simplified form of the expression as rational or irrational. In your final answer, include all of your work.

Answer 1

Given expressions,

`\sqrt[3]{81}`,

`\sqrt[3]{-64}`,

Since,

`\sqrt[3]{81}=(81)^(1)/(3)=(3* 27)^(1)/(3)=3^(1)/(3).(27)^(1)/(3)=3^(1)/(3).(3^3)^(1)/(3)=3(3)^(1)/(3)`

`\sqrt[3]{-64}=(-64)^(1)/(3)=((-4)^3)^(1)/(3)=-4`

Now, a real number is called rational number if it can be expressed as
`(p)/(q)`

Where, p and q are integers,

Such that, q ≠ 0,

Otherwise, the number is called irrational number.

Hence,
`\sqrt[3]{81}`, is irrational number and
`\sqrt[3]{-64}` is a rational number.

Note : ∛3 = irrational number

⇒ 3 ×∛3 = irrational ( Because product of a rational number and an irrational number is always an irrational number. )

Answer 2

`\large\boxed{\sqrt[3]{81}=3\sqrt[3]3-irrational}\\\boxed{\sqrt[3]{-64}=-4-rational}`

Explanation:

`\sqrt[3]{81}=\sqrt[3]{(27)(3)}\qquad\text{use}\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\\\\=\sqrt[3]{27}\cdot\sqrt[3]3=3\sqrt[3]3\qquad/\sqrt[3]{27}=3\ \text{because}\ 3^3=27/\\\\\sqrt[3]{-64}=-4\ \text{because}\ (-4)^3=-64`