By how much does `5sqrt(7) - 2sqrt(5)` exceed `3sqrt(7) - 4sqrt(5)` ?

Options

A. `5(sqrt(7) + sqrt(5))`
B. `sqrt(7) + sqrt(5)`
C. `2(sqrt(7) + sqrt(5))`
D. `7(sqrt(2) + sqrt(5))`

Answer

To solve the problem of how much `5√7 - 2√5` exceeds `3√7 - 4√5`, we will follow these steps:

### Step 1: Set up the expression
We need to find the difference between the two expressions:
\[ (5\sqrt{7} - 2\sqrt{5}) - (3\sqrt{7} - 4\sqrt{5}) \]

### Step 2: Distribute the negative sign
Distributing the negative sign in the second expression gives us:
\[ 5\sqrt{7} - 2\sqrt{5} - 3\sqrt{7} + 4\sqrt{5} \]

### Step 3: Combine like terms
Now we will combine the like terms:
- For the terms involving \(\sqrt{7}\):
\[ 5\sqrt{7} - 3\sqrt{7} = (5 - 3)\sqrt{7} = 2\sqrt{7} \]
- For the terms involving \(\sqrt{5}\):
\[ -2\sqrt{5} + 4\sqrt{5} = (-2 + 4)\sqrt{5} = 2\sqrt{5} \]

### Step 4: Write the final expression
Combining both results, we have:
\[ 2\sqrt{7} + 2\sqrt{5} \]

### Step 5: Factor out the common term
We can factor out the common term \(2\):
\[ 2(\sqrt{7} + \sqrt{5}) \]

### Final Answer
Thus, the amount by which \(5\sqrt{7} - 2\sqrt{5}\) exceeds \(3\sqrt{7} - 4\sqrt{5}\) is:
\[ 2(\sqrt{7} + \sqrt{5}) \]

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