Questions: Rationalize the denominator and simplify. 9/(4√2+2)

Solution Steps

To rationalize the denominator of the expression \( \frac{9}{4\sqrt{2} + 2} \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 4\sqrt{2} - 2 \). This gives us:

\[ \frac{9(4\sqrt{2} - 2)}{(4\sqrt{2} + 2)(4\sqrt{2} - 2)} \]

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

\[ 9(4\sqrt{2} - 2) = 36\sqrt{2} - 18 \]

The denominator simplifies as follows:

\[ (4\sqrt{2})^2 - 2^2 = 32 - 4 = 28 \]

Thus, the expression simplifies to:

\[ \frac{36\sqrt{2} - 18}{28} \]

We can simplify the fraction by dividing both the numerator and the denominator by 2:

\[ \frac{18\sqrt{2} - 9}{14} \]

This can be expressed as:

\[ -\frac{9}{14} + \frac{9\sqrt{2}}{7} \]

Final Answer