Question

What is the product of $\frac{1}{8-3\sqrt{7}}$ and its conjugate?

• A. 1
• B. $8+3\sqrt{7}$
• C. $8-3\sqrt{7}$
• D. $3\sqrt{7}$

Answer 1

The correct option is A

1

So, we must first rationalize the denominator:

$\frac{1}{8-3\sqrt{7}}\times \frac{8+3\sqrt{7}}{8+3\sqrt{7}}$

= $\frac{8+3\sqrt{7}}{8^2-(3\sqrt{7}{)}^2}$

= $\frac{8+3\sqrt{7}}{64-63}$

= $8+3\sqrt{7}$

Now we can find the conjugate by changing the sign in the middle

$8-3\sqrt{7}$

Then, the product of the original number and its conjugate

$\frac{1}{8-3\sqrt{7}}\times$ $8-3\sqrt{7}$

= $(8+3\sqrt{7})\times$ $(8-3\sqrt{7})$

= 64 - 63

= 1