Rationalise the denominator of
$\frac{14}{3+\sqrt{2}}$
Give your answer in its simplest form.
To rationalize the denominator of the expression \(\frac{14}{3+\sqrt{2}}\), we first identify the denominator as \(3+\sqrt{2}\). A common strategy to rationalize denominators containing a square root is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(a + \sqrt{b}\) is \(a - \sqrt{b}\), as this allows us to use the difference of squares identity, \((a+b)(a-b) = a^2 - b^2\), which eliminates the square root.
The conjugate of the denominator \(3+\sqrt{2}\) is \(3-\sqrt{2}\). We multiply the numerator and the denominator of the fraction by this conjugate:
\[\frac{14}{3+\sqrt{2}} \times \frac{3-\sqrt{2}}{3-\sqrt{2}} = \frac{14(3-\sqrt{2})}{(3+\sqrt{2})(3-\sqrt{2})}\]
Now, we expand the denominator using the difference of squares formula:
\[(3+\sqrt{2})(3-\sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7.\]
Next, we expand the numerator:
\[14(3-\sqrt{2}) = 14 \times 3 - 14 \times \sqrt{2} = 42 - 14\sqrt{2}.\]
Thus, the expression becomes:
\[\frac{42 - 14\sqrt{2}}{7}.\]
Finally, we divide each term in the numerator by the denominator:
\[\frac{42 - 14\sqrt{2}}{7} = \frac{42}{7} - \frac{14\sqrt{2}}{7} = 6 - 2\sqrt{2}.\]
The expression is now rationalized, and its simplest form is \(\boxed{6 - 2\sqrt{2}}\).
Rationalizing denominators is a fundamental technique in algebra used to simplify expressions for easier calculation. For example, if you are working with physical measurements or engineering formulas that involve dividing by a radical, simplifying the result into a form like \(A + B\sqrt{C}\) can make it much easier to perform further arithmetic, estimate the value, or plug the result into other equations. This method ensures that the denominator remains a standard integer, facilitating consistency across various mathematical computations.