Evaluate, Correct to one place to decimal, the expression `5/(sqrt(20)-sqrt(10))," if "sqrt(5)=22 and sqrt(10)=3.2`

Answer

To evaluate the expression \( \frac{5}{\sqrt{20} - \sqrt{10}} \) given that \( \sqrt{5} = 2.2 \) and \( \sqrt{10} = 3.2 \), we will follow these steps:

### Step 1: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{20} + \sqrt{10} \).

\[ \frac{5}{\sqrt{20} - \sqrt{10}} \cdot \frac{\sqrt{20} + \sqrt{10}}{\sqrt{20} + \sqrt{10}} = \frac{5(\sqrt{20} + \sqrt{10})}{(\sqrt{20})^2 - (\sqrt{10})^2} \]

### Step 2: Simplify the denominator
Using the identity \( a^2 - b^2 = (a-b)(a+b) \), we can simplify the denominator:

\[ (\sqrt{20})^2 - (\sqrt{10})^2 = 20 - 10 = 10 \]

### Step 3: Substitute values for \( \sqrt{20} \) and \( \sqrt{10} \)
We know that \( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \). Therefore, we can substitute \( \sqrt{5} = 2.2 \):

\[ \sqrt{20} = 2 \cdot 2.2 = 4.4 \]

Now we can substitute \( \sqrt{10} = 3.2 \):

### Step 4: Substitute into the expression
Now we substitute these values back into the expression:

\[ \frac{5(4.4 + 3.2)}{10} \]

### Step 5: Calculate the numerator
Calculate \( 4.4 + 3.2 \):

\[ 4.4 + 3.2 = 7.6 \]

Now substitute this back into the expression:

\[ \frac{5 \cdot 7.6}{10} \]

### Step 6: Calculate the final value
Now multiply \( 5 \cdot 7.6 \):

\[ 5 \cdot 7.6 = 38 \]

Now divide by \( 10 \):

\[ \frac{38}{10} = 3.8 \]

### Final Answer
Thus, the value of the expression \( \frac{5}{\sqrt{20} - \sqrt{10}} \) correct to one decimal place is \( \boxed{3.8} \).

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