Questions: Rationalize the denominator and simplify. 3 / (5 + 2√2)

Solution Steps

To rationalize the denominator of the given expression, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5 + 2\sqrt{2}\) is \(5 - 2\sqrt{2}\). This will eliminate the square root in the denominator.

To rationalize the denominator of the expression \( \frac{3}{5 + 2\sqrt{2}} \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 5 - 2\sqrt{2} \). This gives us:

\[ \frac{3(5 - 2\sqrt{2})}{(5 + 2\sqrt{2})(5 - 2\sqrt{2})} \]

Next, we simplify the numerator and the denominator. The numerator becomes:

\[ 3(5 - 2\sqrt{2}) = 15 - 6\sqrt{2} \]

The denominator simplifies using the difference of squares:

\[ (5 + 2\sqrt{2})(5 - 2\sqrt{2}) = 5^2 - (2\sqrt{2})^2 = 25 - 8 = 17 \]

Thus, the expression simplifies to:

\[ \frac{15 - 6\sqrt{2}}{17} \]

We can separate the fraction into two parts:

\[ \frac{15}{17} - \frac{6\sqrt{2}}{17} \]

Final Answer