If `sqrt(20)+sqrt(125)=15.65`, then what is the value of `sqrt(45)+sqrt(5)` ?

Options

A. `6.98`
B. `8.94`
C. `9.98`
D. `11.27`

Answer

To solve the problem, we need to find the value of \( \sqrt{45} + \sqrt{5} \) given that \( \sqrt{20} + \sqrt{125} = 15.65 \).

**Step 1: Simplify \( \sqrt{20} \) and \( \sqrt{125} \)**

We can simplify \( \sqrt{20} \) and \( \sqrt{125} \) as follows:

\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]

\[ \sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5} \]

**Step 2: Combine the simplified square roots**

Now, we can combine the simplified forms:

\[ \sqrt{20} + \sqrt{125} = 2\sqrt{5} + 5\sqrt{5} = (2 + 5)\sqrt{5} = 7\sqrt{5} \]

**Step 3: Set the equation equal to the given value**

According to the problem, we have:

\[ 7\sqrt{5} = 15.65 \]

**Step 4: Solve for \( \sqrt{5} \)**

Now, we can solve for \( \sqrt{5} \):

\[ \sqrt{5} = \frac{15.65}{7} \]

Calculating this gives:

\[ \sqrt{5} \approx 2.236 \]

**Step 5: Find \( \sqrt{45} + \sqrt{5} \)**

Next, we simplify \( \sqrt{45} \):

\[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \]

Now, we can find \( \sqrt{45} + \sqrt{5} \):

\[ \sqrt{45} + \sqrt{5} = 3\sqrt{5} + \sqrt{5} = (3 + 1)\sqrt{5} = 4\sqrt{5} \]

**Step 6: Substitute the value of \( \sqrt{5} \)**

Substituting \( \sqrt{5} \approx 2.236 \):

\[ 4\sqrt{5} \approx 4 \times 2.236 \approx 8.944 \]

Thus, the value of \( \sqrt{45} + \sqrt{5} \) is approximately \( 8.944 \).

**Final Answer:**
\[ \sqrt{45} + \sqrt{5} \approx 8.944 \]

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