Step 1: Understanding the Expression
To simplify the expression [5(8^(1/3) + 27^(1/3))^3]^(1/4), we start by breaking it down into manageable parts.
Step 2: Calculate the Cube Roots
- 8^(1/3) = 2, because 2 x 2 x 2 = 8.
- 27^(1/3) = 3, because 3 x 3 x 3 = 27.
Now, substitute these values back into the expression:
- (8^(1/3) + 27^(1/3)) = (2 + 3) = 5.
Step 3: Substitute and Simplify
Now the expression simplifies to:
- [5(5)^3]^(1/4).
Calculating (5)^3:
- 5^3 = 125.
So now we have:
- [5 x 125]^(1/4).
Step 4: Multiply
Now calculate:
- 5 x 125 = 625.
Thus, the expression now is:
- [625]^(1/4).
Step 5: Find the Fourth Root
The fourth root of 625 is:
- 625 = 5^4, so (625)^(1/4) = 5.
Final Result
Therefore, the simplified expression is:
- 5.
This process illustrates how to break down complex expressions into simpler components, making calculations easier and more manageable.