Evaluate:
` sqrt ((625)/(729))`

Answer

To evaluate \( \sqrt{\frac{625}{729}} \), we can follow these steps:

### Step 1: Factor the Numerator and Denominator
First, we need to factor both the numerator (625) and the denominator (729).

- **Numerator (625)**:
- \( 625 = 25 \times 25 \)
- \( 25 = 5 \times 5 \)
- Therefore, \( 625 = 5^4 \).

- **Denominator (729)**:
- \( 729 = 27 \times 27 \)
- \( 27 = 3 \times 3 \times 3 \)
- Therefore, \( 729 = 3^6 \).

### Step 2: Rewrite the Expression
Now we can rewrite the expression using the factors we found:

\[ \sqrt{\frac{625}{729}} = \sqrt{\frac{5^4}{3^6}} \]

### Step 3: Apply the Square Root
Using the property of square roots, we can separate the square root of the numerator and the denominator:

\[ \sqrt{\frac{5^4}{3^6}} = \frac{\sqrt{5^4}}{\sqrt{3^6}} \]

### Step 4: Simplify the Square Roots
Now we simplify each square root:

- \( \sqrt{5^4} = 5^{4/2} = 5^2 = 25 \)
- \( \sqrt{3^6} = 3^{6/2} = 3^3 = 27 \)

### Step 5: Combine the Results
Now we can combine the results:

\[ \frac{\sqrt{5^4}}{\sqrt{3^6}} = \frac{25}{27} \]

### Final Answer
Thus, the evaluated result of \( \sqrt{\frac{625}{729}} \) is:

\[ \frac{25}{27} \]

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