Finding the Square Root of \(15 - 4\sqrt {14} \)

The problem asks us to find the square root of the expression \(15 - 4\sqrt {14} \). To do this, we will try to express the number under the square root as a perfect square, specifically in the form \((a-b)^2 = a^2 - 2ab + b^2\).

We need to compare the given expression \(15 - 4\sqrt {14} \) with the form \(a^2 + b^2 - 2ab\).

By comparing, we can see that:

• The constant term is \(a^2 + b^2 = 15\).
• The term with the square root is \(2ab = 4\sqrt {14} \).

From the second equation, \(2ab = 4\sqrt {14} \), we can divide by 2 to get \(ab = 2\sqrt {14} \).

Now, we need to find two numbers \(a\) and \(b\) such that their product is \(2\sqrt {14} \) and the sum of their squares is \(15\).

Let's consider potential pairs for \(a\) and \(b\) whose product is \(2\sqrt {14}\). We can split the terms: \(2 \times \sqrt{14}\). \(\sqrt{14}\) can be split into \(\sqrt{2} \times \sqrt{7}\).

Possible combinations for \(ab\) could be:

• \(a=2\), \(b=\sqrt{14}\): \(a^2+b^2 = 2^2 + (\sqrt{14})^2 = 4 + 14 = 18\). This is not 15.
• \(a=\sqrt{2}\), \(b=2\sqrt{7}\): \(a^2+b^2 = (\sqrt{2})^2 + (2\sqrt{7})^2 = 2 + (4 \times 7) = 2 + 28 = 30\). This is not 15.
• \(a=2\sqrt{2}\), \(b=\sqrt{7}\): \(a^2+b^2 = (2\sqrt{2})^2 + (\sqrt{7})^2 = (4 \times 2) + 7 = 8 + 7 = 15\). This matches the required sum of squares.

So, we have found the numbers: \(a = 2\sqrt{2}\) and \(b = \sqrt{7}\).

Now we can write the expression \(15 - 4\sqrt{14}\) as a perfect square using these values:

Now, we need to find the square root of this expression:

The square root of a perfect square \((x)^2\) is \(|x|\). So, we need to evaluate \(|2\sqrt{2} - \sqrt{7}|\).

We compare \(2\sqrt{2}\) and \(\sqrt{7}\) by squaring them:

• \((2\sqrt{2})^2 = 4 \times 2 = 8\)
• \((\sqrt{7})^2 = 7\)

Since \(8 > 7\), it means \(2\sqrt{2} > \sqrt{7}\). Therefore, the expression \(2\sqrt{2} - \sqrt{7}\) is positive.

Thus, the square root is:

The square root of \(15 - 4\sqrt{14}\) is \(2\sqrt{2} - \sqrt{7}\).

Revision Table: Key Concepts

Additional Information: Working with Surds

Surds are expressions containing irrational square roots, like \(\sqrt{2}\) or \(\sqrt{7}\). Simplifying expressions involving surds often requires techniques similar to algebraic manipulation.

One common technique is to recognize or create a perfect square under the square root sign, as we did in this problem. This is particularly useful for expressions of the form \(a \pm \sqrt{b}\) or \(a \pm c\sqrt{d}\).

For an expression \(\sqrt{X \pm Y\sqrt{Z}}\), we often look for two numbers whose sum is \(X\) and whose product is related to \(Y^2 Z\). In our problem, \(X=15\) and \(Y\sqrt{Z} = 4\sqrt{14}\). So we looked for \(a, b\) such that \(a^2+b^2 = 15\) and \(2ab = 4\sqrt{14}\).

This method is a standard approach for denesting square roots of this form.