Evaluate:
`sqrt (1.69)`

Options

A. `1.2`
B. `1.4`
C. `1.5`
D. `1.3`

Answer

To evaluate \( \sqrt{1.69} \), we can use two methods: prime factorization and the long division method. Let's go through both methods step by step.

### Method 1: Prime Factorization

1. **Remove the Decimal**:
To make it easier to work with, we can remove the decimal by multiplying both the numerator and the denominator by 100. Thus, we have:
\[ 1.69 = \frac{169}{100} \]

2. **Take the Square Root**:
Now we can apply the square root to both the numerator and the denominator:
\[ \sqrt{1.69} = \sqrt{\frac{169}{100}} = \frac{\sqrt{169}}{\sqrt{100}} \]

3. **Calculate the Square Roots**:
We know that:
\[ \sqrt{169} = 13 \quad \text{and} \quad \sqrt{100} = 10 \]
Therefore:
\[ \sqrt{1.69} = \frac{13}{10} \]

4. **Convert to Decimal**:
Dividing 13 by 10 gives us:
\[ \frac{13}{10} = 1.3 \]

### Conclusion for Method 1:
Thus, \( \sqrt{1.69} = 1.3 \).

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### Method 2: Long Division Method

1. **Set Up the Long Division**:
Write 1.69 as 169 under the square root, and set up the long division method. Since we are dealing with decimals, we will consider two decimal places.

2. **Pair the Digits**:
Starting from the decimal point, pair the digits: (1)(69).

3. **Find the Largest Square**:
Find the largest square less than or equal to 1. The largest square is \( 1^2 = 1 \). Write 1 above the line.

4. **Subtract and Bring Down**:
Subtract \( 1 \) from \( 1 \) to get \( 0 \), then bring down the next pair (69) to get 69.

5. **Double the Root**:
Double the number above the line (which is 1) to get 2. Now we need to find a digit \( x \) such that \( 2x \times x \) is less than or equal to 69.

6. **Find the Digit**:
Testing \( x = 3 \):
\[ 23 \times 3 = 69 \]
This works perfectly. Write 3 above the line next to 1.

7. **Final Result**:
The result from the long division method gives us:
\[ \sqrt{1.69} = 1.3 \]

### Conclusion for Method 2:
Thus, using the long division method, we also find that \( \sqrt{1.69} = 1.3 \).

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### Final Answer:
\[ \sqrt{1.69} = 1.3 \]

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