What is the irrational cube root 3^√81 rewritten as a product of an integer and another irrational cube root?
We can rewrite 81 as a cube of another number, 3^4 = 81. Therefore, we can rewrite the expression as ∛(3^√81) = ∛(3^4) = (3^4)^1/3.
Using the property of exponents, we can rewrite this as (3^1/3) * (3^4/3).
Therefore, the irrational cube root 3^√81 can be rewritten as a product of an integer and another irrational cube root, which is 3^1/3.
To rewrite the irrational cube root 3^√81 as a product of an integer and another irrational cube root, we can simplify it as follows:
First, let's express 81 as a perfect cube. We can write 81 as 3^4 since 3^4 = 81.
Now we can rewrite the original expression 3^√81 as (3^4)^√3. According to the laws of exponents, raising a number to an exponent divided by a whole number is equivalent to taking the whole number root of the number first and then raising it to the exponent. Therefore, we have:
(3^4)^√3 = 3^(4*√3)
Simplifying further, we can rewrite it as:
3^(4*√3) = 3^(2√3*2)
Breaking it down, we have:
3^(2√3*2) = (3^(√3))^2
Therefore, the irrational cube root 3^√81 can be rewritten as an integer (3^(√3))^2 multiplied by the irrational cube root of 3.
To rewrite the irrational cube root ∛81 as a product of an integer and another irrational cube root, we need to find the largest perfect cube that divides 81.
To do this, we can start by finding the prime factorization of 81:
81 = 3 * 3 * 3 * 3 = 3^4
Now, we can rewrite 81 as:
81 = (3^4) * 1
Taking the cube root of both sides, we get:
∛81 = ∛((3^4) * 1)
Using the property of the cube root that states ∛(a * b) = ∛a * ∛b, we can split the cube root into two factors:
∛81 = ∛(3^4) * ∛1
Simplifying the cube root of the perfect cube, we have:
∛81 = 3 * ∛1
Since 3 is an integer and ∛1 is an irrational cube root, we have successfully rewritten √81 as a product of an integer and another irrational cube root:
∛81 = 3 * ∛1