Evaluate:
`(sqrt 80)/(sqrt(405))`

Answer

To evaluate the expression \(\frac{\sqrt{80}}{\sqrt{405}}\), we will use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 80
First, we need to find the prime factors of 80.

- Divide 80 by the smallest prime number, which is 2:
\[ 80 \div 2 = 40 \]
- Divide 40 by 2:
\[ 40 \div 2 = 20 \]
- Divide 20 by 2:
\[ 20 \div 2 = 10 \]
- Divide 10 by 2:
\[ 10 \div 2 = 5 \]
- Finally, 5 is a prime number, and we stop here.

So, the prime factorization of 80 is:
\[ 80 = 2^4 \times 5^1 \]

### Step 2: Prime Factorization of 405
Next, we find the prime factors of 405.

- 405 is not divisible by 2 (it is odd), so we try the next smallest prime number, which is 3:
\[ 405 \div 3 = 135 \]
- Divide 135 by 3:
\[ 135 \div 3 = 45 \]
- Divide 45 by 3:
\[ 45 \div 3 = 15 \]
- Divide 15 by 3:
\[ 15 \div 3 = 5 \]
- Finally, 5 is a prime number, and we stop here.

So, the prime factorization of 405 is:
\[ 405 = 3^4 \times 5^1 \]

### Step 3: Substitute the Prime Factorizations
Now we can rewrite the original expression using the prime factorizations we found:
\[ \frac{\sqrt{80}}{\sqrt{405}} = \frac{\sqrt{2^4 \times 5^1}}{\sqrt{3^4 \times 5^1}} \]

### Step 4: Simplify the Square Roots
Using the property of square roots, we can simplify:
\[ \frac{\sqrt{2^4} \times \sqrt{5^1}}{\sqrt{3^4} \times \sqrt{5^1}} = \frac{2^2 \times \sqrt{5}}{3^2 \times \sqrt{5}} \]

### Step 5: Cancel the Common Terms
Notice that \(\sqrt{5}\) appears in both the numerator and the denominator:
\[ = \frac{2^2}{3^2} = \frac{4}{9} \]

### Final Answer
Thus, the evaluated expression is:
\[ \frac{\sqrt{80}}{\sqrt{405}} = \frac{4}{9} \]

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