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^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.

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Answer:

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

Step-by-step explanation:

[tex]\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[/tex]

Answer:

Given expressions,

[tex]\sqrt[3]{81}[/tex],

[tex]\sqrt[3]{-64}[/tex],

Since,

[tex]\sqrt[3]{81}=(81)^\frac{1}{3}=(3\times 27)^\frac{1}{3}=3^\frac{1}{3}.(27)^\frac{1}{3}=3^\frac{1}{3}.(3^3)^\frac{1}{3}=3(3)^\frac{1}{3}[/tex]

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

Now, a real number is called rational number if it can be expressed as [tex]\frac{p}{q}[/tex]

Where, p and q are integers,

Such that, q ≠ 0,

Otherwise, the number is called irrational number.

Hence, [tex]\sqrt[3]{81}[/tex], is irrational number and [tex]\sqrt[3]{-64}[/tex] 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. )

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