3ab^(3)(\root(3)(2ab^(2)))
To simplify the expression 3ab^3(root(3)(2ab^2)), we can start by simplifying the cube root.
The cube root of (2ab^2) is written as (2ab^2)^(1/3). This means we need to find a value that, when cubed, equals 2ab^2.
Now, let's simplify the expression inside the parentheses: 2ab^2^(1/3).
The value of 2ab^2^(1/3) is simply the cube root of (2ab^2), which is (2ab^2)^(1/3).
So, the simplified expression is 3ab^3(2ab^(2/3)).
We can also simplify this further by multiplying the coefficients and combining like terms, as follows:
3 * 2 = 6
ab^3 * ab^(2/3) = a^1 * a^1 * a^1 * b^(3+2/3) = a^3 * b^(11/3)
Thus, the simplified expression is 6a^3b^(11/3).
To simplify the expression 3ab^3 * (∛(2ab^2)), we can begin by looking at each term separately.
First, let's look at the expression 3ab^3:
- The coefficient 3 remains unchanged because there is no opportunity to simplify it further.
- In the variable term ab^3, we can multiply the coefficient a with ∛(2a) to simplify it.
Now, let's focus on the expression inside the cube root (∛(2ab^2)):
- Since the cube root (∛) and exponent (^) are inverse operations, we can simplify b^2 to ∛(b^6).
Putting it all together, we have:
3ab^3 * (∛(2ab^2)) = 3a * (∛(2a)) * (∛(b^6))
Now, because the cube root (∛) and exponent (^) are inverse operations, (∛(b^6)) simplifies to b^(6/3), which gives us b^2.
Therefore, the simplified expression is:
3ab^3 * (∛(2ab^2)) = 3a * (∛(2a)) * (∛(b^6)) = 3a * (∛(2a)) * b^2
To simplify the expression 3ab^3 * √(2ab^2), we can break it down into multiple steps:
Step 1: Simplify the square root term.
√(2ab^2) = (√2) * (√(ab^2))
= (√2) * (b * √a)
Step 2: Multiply the terms.
3ab^3 * √(2ab^2) = 3ab^3 * (√2) * (b * √a)
= 3 * b * b^3 * √2 * √a
= 3 * b^(1+3) * √(2a)
= 3b^4 * √(2a)
Therefore, 3ab^3 * √(2ab^2) simplifies to 3b^4 * √(2a).