Roots and Irrational Numbers.
Rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root. Show your work.
To rewrite ^3√189 as the product of an integer and another irrational cube root, we need to find a perfect cube that is a factor of 189.
Let's start by finding the prime factorization of 189:
189 = 3 * 63
= 3 * 3 * 21
= 3 * 3 * 3 * 7
From the prime factorization, we can see that 189 is divisible by 3 three times.
Now, let's rewrite the cube root of 189:
^3√189 = ^3√(3 * 3 * 3 * 7)
We can take out one of the perfect cubes:
^3√(3 * 3 * 3 * 7) = ^3√(3^3 * 7)
This can be rewritten as:
^3√(3^3) * ^3√7
So, ^3√189 is equivalent to 3 * ^3√7.
To rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root, we need to find a perfect cube that divides evenly into 189.
Let's start by finding the prime factorization of 189:
189 = 3 * 63
= 3 * 3 * 21
= 3 * 3 * 3 * 7
Now, we can see that 189 has a perfect cube factor of 27 (which is 3^3). So, we can rewrite 189 as:
189 = 27 * 7
Therefore, we can rewrite ^3√189 as:
^3√(27 * 7)
Since ^3√27 is an integer (equal to 3), we can write it as 3 times another irrational cube root:
^3√189 = 3 * ^3√7
Hence, the irrational cube root ^3√189 can be written as the product of an integer (3) and another irrational cube root (^3√7).
To rewrite the irrational cube root ^3√189 as the product of an integer and another irrational cube root, we look for perfect cube factors of 189.
First, let's find the prime factorization of 189:
189 = 3 × 63
= 3 × 3 × 21
= 3 × 3 × 3 × 7
We see that 189 has a perfect cube factor of 3^3 = 27. We can write 189 as the product of 27 and another factor that is not a perfect cube.
189 = 27 × 7
Now, let's rewrite the original expression:
^3√189 = ^3√(27 × 7)
According to the properties of cube roots, we can separate the product inside the cube root into individual cube roots:
^3√(27 × 7) = ^3√27 × ^3√7
Since 27 is a perfect cube (3^3), we can simplify ^3√27 as 3. Therefore, we have:
^3√189 = 3 × ^3√7
Hence, ^3√189 can be rewritten as the product of an integer (3) and another irrational cube root (√7).