Estimate : `sqrt(60)`

Options

A. 7.7
B. 6.9
C. 7.1
D. 4.46

Answer

To estimate \(\sqrt{60}\), we can follow these steps:

### Step 1: Prime Factorization
First, we need to factor 60 into its prime components. We can express 60 as:
\[ 60 = 6 \times 10 \]
Next, we can break down 6 and 10 into their prime factors:
\[ 6 = 2 \times 3 \quad \text{and} \quad 10 = 2 \times 5 \]
So, we can rewrite 60 as:
\[ 60 = 2 \times 3 \times 2 \times 5 = 2^2 \times 3 \times 5 \]

### Step 2: Rewrite the Square Root
Now, we can express \(\sqrt{60}\) using the prime factorization:
\[ \sqrt{60} = \sqrt{2^2 \times 3 \times 5} \]

### Step 3: Simplify the Square Root
Using the property of square roots, we can separate the square root of the perfect square:
\[ \sqrt{60} = \sqrt{2^2} \times \sqrt{3} \times \sqrt{5} \]
Since \(\sqrt{2^2} = 2\), we have:
\[ \sqrt{60} = 2 \times \sqrt{3} \times \sqrt{5} \]

### Step 4: Estimate \(\sqrt{3}\) and \(\sqrt{5}\)
Next, we need to estimate \(\sqrt{3}\) and \(\sqrt{5}\):
- \(\sqrt{3} \approx 1.73\)
- \(\sqrt{5} \approx 2.23\)

### Step 5: Multiply the Estimates
Now we can substitute these estimates back into our expression:
\[ \sqrt{60} \approx 2 \times 1.73 \times 2.23 \]
Calculating this step-by-step:
1. First, calculate \(2 \times 1.73\):
\[ 2 \times 1.73 = 3.46 \]
2. Next, multiply \(3.46\) by \(2.23\):
\[ 3.46 \times 2.23 \approx 7.7 \]

### Conclusion
Thus, the estimated value of \(\sqrt{60}\) is approximately:
\[ \sqrt{60} \approx 7.7 \]