Question

Please solve and explain the surdic equation:
Rationalize the Expression
(7-√3/13-√3) * (13+√3/13+√3)

Answer 1

To rationalize the expression \(\frac{7 - \sqrt{3}}{13 - \sqrt{3}}\) by multiplying by the conjugate of the denominator, we can follow these steps:

1. **Identify the Conjugate**: The conjugate of \(13 - \sqrt{3}\) is \(13 + \sqrt{3}\).

2. **Multiply the Numerator and Denominator**: We can multiply both the numerator and denominator by the conjugate:

\[
\frac{(7 - \sqrt{3})(13 + \sqrt{3})}{(13 - \sqrt{3})(13 + \sqrt{3})}
\]

3. **Simplify the Denominator**: Using the difference of squares, we can simplify the denominator:

\[
(13 - \sqrt{3})(13 + \sqrt{3}) = 13^2 - (\sqrt{3})^2 = 169 - 3 = 166
\]

4. **Simplify the Numerator**: Now for the numerator:

\[
(7 - \sqrt{3})(13 + \sqrt{3}) = 7 \cdot 13 + 7 \cdot \sqrt{3} - \sqrt{3} \cdot 13 - (\sqrt{3})^2
\]
\[
= 91 + 7\sqrt{3} - 13\sqrt{3} - 3 = 91 - 3 + (7\sqrt{3} - 13\sqrt{3})
\]
\[
= 88 - 6\sqrt{3}
\]

5. **Put It All Together**: The entire expression now looks like this:

\[
\frac{88 - 6\sqrt{3}}{166}
\]

6. **Simplify the Fraction**: We can factor out a common factor if necessary:

\[
= \frac{88}{166} - \frac{6\sqrt{3}}{166}
\]

Let's simplify \(\frac{88}{166}\):

\[
\frac{88}{166} = \frac{44}{83}
\]

So the final result can be expressed as:

\[
\frac{44}{83} - \frac{6\sqrt{3}}{166}
\]

Thus, the rationalized form of the expression \(\frac{7 - \sqrt{3}}{13 - \sqrt{3}}\) is:

\[
\frac{44}{83} - \frac{6\sqrt{3}}{166}
\]