Rewrite the irrational cube root 3 square root 189 was the product of an integer and another irrational cube root. Show your work
To rewrite the irrational cube root of 3 multiplied by the square root of 189 as the product of an integer and another irrational cube root, we can simplify the expression and use prime factorization.
First, let's express 189 as the product of its prime factors:
189 = 3 * 3 * 3 * 7
Next, we can simplify the square root of 189:
√189 = √(3 * 3 * 3 * 7) = 3√7
Therefore, the expression can be simplified as:
3^(1/3) * √189 = 3^(1/3) * 3√7 = 3^(1/3) * 3^(1/2) * √7 = (3^(1/3 + 1/2)) * √7
Now, let's find the rational number that is equivalent to 3^(1/3 + 1/2) by finding a common denominator:
1/3 + 1/2 = 2/6 + 3/6 = 5/6
Therefore, 3^(1/3 + 1/2) is equivalent to 3^(5/6).
Hence, the irrational cube root of 3 multiplied by the square root of 189 can be expressed as the product of an integer and another irrational cube root:
3^(1/3) * √189 = 3^(1/3 + 1/2) * √7 = 3^(5/6) * √7