rewrite the irrational cube root of ^3R189 as the product of an integer and another irrational cube root.
∛189 can be written as ∛(27*7), which can further be simplified as 3∛7.
Therefore, the irrational cube root of ^3R189 can be written as 3∛7.
To rewrite the irrational cube root of ^3√189 as the product of an integer and another irrational cube root, we need to find a perfect cube factor of 189.
Let's find the prime factorization of 189:
189 = 3 * 3 * 3 * 7
Now, we can write 189 as the product of two terms:
^3√189 = ^3√(3 * 3 * 3 * 7)
We can rearrange the terms to group a perfect cube factor together:
^3√189 = (^3√(3 * 3 * 3)) * ^3√7
Simplifying, we can write the final answer as:
^3√189 = 3 * ^3√7
Therefore, the irrational cube root of ^3√189 can be rewritten as the product of an integer (3) and another irrational cube root (√7).
To rewrite the irrational cube root of ^3√189 as the product of an integer and another irrational cube root, we need to find a perfect cube factor of 189 and extract it from the radical.
Step 1: Prime Factorization
Find the prime factorization of 189. We have:
189 = 3 * 3 * 3 * 7
Step 2: Identify Perfect Cube Factors
Look for perfect cube factors among these prime factors. In this case, we have one perfect cube factor: 3 * 3 * 3 = 27.
Step 3: Rewrite the Radical
Now, we can rewrite the cube root of ^3√189 as the product of 27 and another radical containing the remaining factors (7 in this case). Let's call this remaining factor "x":
^3√189 = ^3√(27 * 7)
Thus, we can rewrite it as:
^3√189 = 27 * ^3√7
Therefore, the irrational cube root of ^3√189 can be rewritten as the product of an integer (27) and another irrational cube root (^3√7).