Find the values of a and b in each of the
`(sqrt(7)-2)/(sqrt(7)+2)= a sqrt(7)+b`

Answer

To find the values of \( a \) and \( b \) in the equation

\[ \frac{\sqrt{7}-2}{\sqrt{7}+2} = a\sqrt{7} + b, \]

we will follow these steps:

### Step 1: Multiply by the Conjugate
To simplify the left-hand side, we will multiply both the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{7} - 2 \):

\[ \frac{(\sqrt{7}-2)(\sqrt{7}-2)}{(\sqrt{7}+2)(\sqrt{7}-2)}. \]

### Step 2: Expand the Numerator and Denominator
Now, we will expand both the numerator and the denominator:

- **Numerator**:
\[ (\sqrt{7}-2)(\sqrt{7}-2) = (\sqrt{7})^2 - 2 \cdot \sqrt{7} \cdot 2 + 2^2 = 7 - 4\sqrt{7} + 4 = 11 - 4\sqrt{7}. \]

- **Denominator**:
\[ (\sqrt{7}+2)(\sqrt{7}-2) = (\sqrt{7})^2 - (2)^2 = 7 - 4 = 3. \]

So, we have:

\[ \frac{11 - 4\sqrt{7}}{3}. \]

### Step 3: Simplify the Expression
Now we can simplify the expression:

\[ \frac{11}{3} - \frac{4\sqrt{7}}{3}. \]

This can be rewritten as:

\[ -\frac{4}{3}\sqrt{7} + \frac{11}{3}. \]

### Step 4: Compare with the Right Side
Now we compare this with the right side of the equation \( a\sqrt{7} + b \):

\[ -\frac{4}{3}\sqrt{7} + \frac{11}{3} = a\sqrt{7} + b. \]

From this comparison, we can identify:

- The coefficient of \( \sqrt{7} \): \( a = -\frac{4}{3} \)
- The constant term: \( b = \frac{11}{3} \)

### Final Values
Thus, the values of \( a \) and \( b \) are:

\[ a = -\frac{4}{3}, \quad b = \frac{11}{3}. \]

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