Rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root. Show your work.
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(2 points)
To rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root, we need to factor 189 and look for perfect cube factors.
The prime factorization of 189 is 3 * 3 * 3 * 7.
Let's separate the perfect cube factor (3 * 3 * 3) from the remaining prime factor (7):
^3√189 = ^3√(3 * 3 * 3 * 7)
Now, rewrite it as:
^3√189 = ^3√(27 * 7)
The perfect cube factor 27 can be written as 3 * 3 * 3:
^3√(27 * 7) = ^3√(3 * 3 * 3 * 7)
Now, we can rewrite it as the product of an integer (3) and another irrational cube root:
^3√(3 * 3 * 3 * 7) = 3 * ^3√7
Therefore, ^3√189 can be written as the product of the integer 3 and the irrational cube root ^3√7.