`sqrt(27^(2/3)+128 ^(4/7))=`

Answer

To solve the expression \( \sqrt{27^{2/3} + 128^{4/7}} \), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers
First, we can express 27 and 128 in terms of their prime bases:
- \( 27 = 3^3 \)
- \( 128 = 2^7 \)

So we can rewrite the expression as:
\[ \sqrt{(3^3)^{2/3} + (2^7)^{4/7}} \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify each term:
\[ (3^3)^{2/3} = 3^{3 \cdot (2/3)} = 3^2 \]
\[ (2^7)^{4/7} = 2^{7 \cdot (4/7)} = 2^4 \]

Now, substituting these back into the expression gives us:
\[ \sqrt{3^2 + 2^4} \]

### Step 3: Calculate the values
Next, we calculate the values of \( 3^2 \) and \( 2^4 \):
\[ 3^2 = 9 \]
\[ 2^4 = 16 \]

### Step 4: Add the results
Now we add these two results together:
\[ 9 + 16 = 25 \]

### Step 5: Take the square root
Finally, we take the square root of the sum:
\[ \sqrt{25} = 5 \]

### Final Answer
Thus, the value of \( \sqrt{27^{2/3} + 128^{4/7}} \) is \( 5 \).

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