Question

Rewrite the irrational cube root 3^√189 as the product of an integer and another irrational cube root. Show your work.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To rewrite the irrational cube root $\sqrt[3]{189}$ as the product of an integer and another irrational cube root, let's start by finding the integer cube closest to $\sqrt[3]{189}$.

Since $5^3=125$ and $6^3=216$, the closest integer cubes to $189$ are $125$ and $216$.

To decide which integer cube is closest to $189$, we can calculate the differences:
$189 - 125 = 64$ and $216 - 189 = 27$.

Since $64 > 27$, the cube root of $189$ is closer to $5^3=125$ than $6^3=216$.

Now we can rewrite the cube root of $189$ as follows:

$\sqrt[3]{189} = \sqrt[3]{125 \cdot \frac{189}{125}}$

Simplifying the numerator: $189 = 125 \cdot \frac{189}{125} = 125 \cdot \frac{3^3}{5^3} = (3 \cdot 5) \cdot \frac{(3 \cdot 3 \cdot 3)}{(5 \cdot 5 \cdot 5)}$

Now, rewrite it as the product of an integer and another irrational cube root:

$\sqrt[3]{189} = \sqrt[3]{125} \cdot \sqrt[3]{\frac{27}{125}}$

Therefore, the irrational cube root $\sqrt[3]{189}$ can be rewritten as the product of the integer cube root $\sqrt[3]{125}$ and the irrational cube root $\sqrt[3]{\frac{27}{125}}$.