Rewrite the irrational cube root ^3√189
as the product of an integer and another irrational cube root. Show your work.
We can rewrite the number 189 as the product of a perfect cube and another positive integer.
The nearest perfect cube to 189 is 125 (which is equal to 5^3).
So, we can rewrite 189 as:
189 = 125 * 1.512
Now, we can write the original expression ^3√189 as:
^3√189 = ^3√(125 * 1.512)
Since the cube root of 125 is 5, we can simplify the expression as:
^3√(125 * 1.512) = 5 * ^3√1.512
Thus, the irrational cube root ^3√189 can be written as the product of an integer (5) and another irrational cube root (√1.512).
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 factor of 189.
Let's analyze the prime factorization of 189:
189 = 3 * 3 * 3 * 7
From the prime factorization, we can see that 189 contains a perfect cube factor, which is 3 * 3 * 3 = 27.
Now, we can rewrite ^3√189 as ^3√(27 * 7).
Using the property of cube roots, we can split the cube root into two separate cube roots:
^3√(27 * 7) = ^3√27 * ^3√7
Simplifying further, ^3√27 is an integer since it equals 3:
^3√27 = 3
Therefore, we can rewrite ^3√189 as 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 has a factor of 189.
Let's list the cubes of numbers until we find one that divides 189:
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
6^3 = 216
None of these perfect cubes divide 189 completely. However, 6^3 is the closest to 189. Let's divide 189 by 216:
189 ÷ 216 = 0.875
Since none of the perfect cubes divide 189, we can't simplify ^3√189 any further. Therefore, we can rewrite ^3√189 as:
^3√189 = 6 * ^3√0.875
Hence, the irrational cube root ^3√189 can be written as the product of an integer (6) and another irrational cube root (^3√0.875).