rewrite the irrational cube root 3^ to the square root of 189 as the product of an integer and another irrational cube root. show your work
To rewrite the irrational cube root of 3√ as the square root of 189, we need to express it as the product of an integer and another irrational cube root.
Let's first simplify the expression √189. We can express 189 as the product of a perfect square and a prime number: 189 = 9 * 21.
Taking the square root, we have: √189 = √(9 * 21).
Since the square root of 9 is a rational number (integer), we can simplify it further: √189 = 3√21.
Therefore, the irrational cube root of 3√ can be rewritten as the product of an integer (3) and another irrational cube root (√21).
√189 = 3√21
but they were asking about cube roots:
∛189 = ∛(27*7) = 3∛7
Apologies for the oversight. You are correct.
To rewrite the irrational cube root of 3√ as the square root of 189, we need to express it as the product of an integer and another irrational cube root.
Let's begin by finding the prime factorization of 189: 189 = 27 * 7.
Now, we can rewrite it as the product of an integer (3) and another irrational cube root (∛7): ∛189 = 3∛7.
Therefore, the irrational cube root of 3√ can be rewritten as the product of an integer (3) and another irrational cube root (∛7).
To rewrite the irrational cube root of 3 as the square root of 189 as the product of an integer and another irrational cube root, we can utilize the property that the cube root of a number is equal to the number raised to the power of 1/3.
First, let's express the irrational cube root of 3 as 3^(1/3). Now, we need to rewrite the square root of 189 as a cube root.
To do this, we can express the square root of 189 as (189)^(1/2). Now we want to find a perfect cube factor of 189, since that will allow us to express it as a rational number multiplied by another irrational cube root.
The prime factorization of 189 is 3 * 3 * 3 * 7. We can see that 189 can be expressed as (3 * 3 * 3) * (3 * 7).
So, we can rewrite the square root of 189 as the product of two factors: the cube root of (3 * 3 * 3) and the cube root of (3 * 7).
Therefore, the square root of 189 can be expressed as (3^3)^(1/3) * (3 * 7)^(1/3).
Simplifying this expression, we have 3^(3 * 1/3) * (3 * 7)^(1/3), which can be further simplified to 3 * 3^(1/3) * (3 * 7)^(1/3).
So, the irrational cube root of 3^ can be rewritten as the product of an integer (3) and another irrational cube root (3 * 7)^(1/3).
To summarize:
∛3^ = 3 * (3 * 7)^(1/3)