Simplifying Radical Expressions and Finding Ratios

The problem asks us to simplify a given mathematical expression involving radicals, express it in the form \(a + \sqrt 5 b\), and then find the value of the ratio \((ab) \div (a + b)\).

The given expression is:

\[ \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3\, + \,\sqrt 5 }}} \]

We need to simplify the denominator first. The term \(\sqrt {3\, + \,\sqrt 5 }\) is a nested radical. We can simplify this type of radical if the expression inside the square root can be written in the form \((\sqrt x + \sqrt y)^2\) or \((\sqrt x - \sqrt y)^2\). For \(\sqrt{A \pm \sqrt B}\), we look for \(x, y\) such that \(x+y = A\) and \(4xy = B\). In our case, \(A=3\) and \(B=5\). So we need \(x+y = 3\) and \(4xy = 5\), which means \(xy = \frac{5}{4}\). We need two numbers that add up to 3 and multiply to \(\frac{5}{4}\).

Consider the quadratic equation \(t^2 - (x+y)t + xy = 0\), which is \(t^2 - 3t + \frac{5}{4} = 0\). Multiplying by 4, we get \(4t^2 - 12t + 5 = 0\). Using the quadratic formula:

\[ t = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(5)}}{2(4)} = \frac{12 \pm \sqrt{144 - 80}}{8} = \frac{12 \pm \sqrt{64}}{8} = \frac{12 \pm 8}{8} \]

The two possible values for \(t\) are \(\frac{12+8}{8} = \frac{20}{8} = \frac{5}{2}\) and \(\frac{12-8}{8} = \frac{4}{8} = \frac{1}{2}\). So, we can take \(x = \frac{5}{2}\) and \(y = \frac{1}{2}\).

Thus, \(\sqrt {3\, + \,\sqrt 5 } = \sqrt{\frac{5}{2} + \frac{1}{2} + 2\sqrt{\frac{5}{2} \cdot \frac{1}{2}}} = \sqrt{\left(\sqrt{\frac{5}{2}} + \sqrt{\frac{1}{2}}\right)^2} = \sqrt{\frac{5}{2}} + \sqrt{\frac{1}{2}}\). To simplify, we can rationalize the denominators within the nested radical terms:

\[ \sqrt{\frac{5}{2}} = \frac{\sqrt 5}{\sqrt 2} = \frac{\sqrt 5 \cdot \sqrt 2}{\sqrt 2 \cdot \sqrt 2} = \frac{\sqrt{10}}{2} \]

\[ \sqrt{\frac{1}{2}} = \frac{1}{\sqrt 2} = \frac{1 \cdot \sqrt 2}{\sqrt 2 \cdot \sqrt 2} = \frac{\sqrt 2}{2} \]

Alternatively, we can keep the \(\sqrt 2\) in the denominator and simplify later. \[ \sqrt {3\, + \,\sqrt 5 } = \sqrt{\frac{5}{2}} + \sqrt{\frac{1}{2}} = \frac{\sqrt 5}{\sqrt 2} + \frac{1}{\sqrt 2} = \frac{\sqrt 5 + 1}{\sqrt 2} \] Now substitute this back into the denominator of the original expression:

\[ 4\sqrt 2 - \sqrt {3\, + \,\sqrt 5 } = 4\sqrt 2 - \frac{\sqrt 5 + 1}{\sqrt 2} \] To combine these terms, find a common denominator, which is \(\sqrt 2\):

\[ 4\sqrt 2 - \frac{\sqrt 5 + 1}{\sqrt 2} = \frac{4\sqrt 2 \cdot \sqrt 2}{\sqrt 2} - \frac{\sqrt 5 + 1}{\sqrt 2} = \frac{4(2) - (\sqrt 5 + 1)}{\sqrt 2} = \frac{8 - \sqrt 5 - 1}{\sqrt 2} = \frac{7 - \sqrt 5}{\sqrt 2} \]

Now substitute this simplified denominator back into the original expression:

\[ \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3\, + \,\sqrt 5 }}} = \frac{{22\sqrt 2 }}{{\frac{7 - \sqrt 5}{\sqrt 2}}} \]

When dividing by a fraction, we multiply by its reciprocal:

\[ 22\sqrt 2 \cdot \frac{\sqrt 2}{7 - \sqrt 5} = \frac{22 \cdot (\sqrt 2)^2}{7 - \sqrt 5} = \frac{22 \cdot 2}{7 - \sqrt 5} = \frac{44}{7 - \sqrt 5} \]

To express this in the form \(a + \sqrt 5 b\), we need to rationalize the denominator. Multiply the numerator and the denominator by the conjugate of \(7 - \sqrt 5\), which is \(7 + \sqrt 5\):

\[ \frac{44}{7 - \sqrt 5} \cdot \frac{7 + \sqrt 5}{7 + \sqrt 5} = \frac{44(7 + \sqrt 5)}{(7 - \sqrt 5)(7 + \sqrt 5)} \]

Using the difference of squares formula, \((x-y)(x+y) = x^2 - y^2\), the denominator becomes:

\[ (7 - \sqrt 5)(7 + \sqrt 5) = 7^2 - (\sqrt 5)^2 = 49 - 5 = 44 \]

So the expression simplifies to:

\[ \frac{44(7 + \sqrt 5)}{44} = 7 + \sqrt 5 \]

We are given that the expression equals \(a + \sqrt 5 b\), with \(a, b > 0\). Comparing \(7 + \sqrt 5\) to \(a + \sqrt 5 b\), we get \(a = 7\) and \(b = 1\). Both \(a\) and \(b\) are indeed positive.

Now we need to find the value of \((ab) \div (a + b)\).

Calculate \(ab\):

\[ ab = 7 \cdot 1 = 7 \]

Calculate \(a + b\):

\[ a + b = 7 + 1 = 8 \]

Calculate the ratio \((ab) \div (a + b)\):

\[ \frac{ab}{a+b} = \frac{7}{8} \]

This ratio can be expressed as 7 ∶ 8.

Let's verify the steps and the final result.

• Simplified nested radical: \(\sqrt{3+\sqrt{5}} = \frac{1+\sqrt{5}}{\sqrt{2}}\). Correct.
• Simplified denominator: \(4\sqrt{2} - \frac{1+\sqrt{5}}{\sqrt{2}} = \frac{8 - (1+\sqrt{5})}{\sqrt{2}} = \frac{7-\sqrt{5}}{\sqrt{2}}\). Correct.
• Overall expression: \(\frac{22\sqrt{2}}{(7-\sqrt{5})/\sqrt{2}} = \frac{22\sqrt{2} \cdot \sqrt{2}}{7-\sqrt{5}} = \frac{44}{7-\sqrt{5}}\). Correct.
• Rationalized expression: \(\frac{44}{7-\sqrt{5}} \cdot \frac{7+\sqrt{5}}{7+\sqrt{5}} = \frac{44(7+\sqrt{5})}{49-5} = \frac{44(7+\sqrt{5})}{44} = 7+\sqrt{5}\). Correct.
• Identified \(a=7, b=1\). Correct, and both are positive.
• Calculated \(ab = 7 \cdot 1 = 7\). Correct.
• Calculated \(a+b = 7 + 1 = 8\). Correct.
• Calculated ratio \(\frac{ab}{a+b} = \frac{7}{8}\). Correct.

The final ratio is 7 ∶ 8.

Final Answer is 7 ∶ 8.

Revision Table: Key Concepts for Simplifying Radicals

Additional Information on Radical Expressions and Ratios

Radical expressions are expressions that include a radical symbol (\(\sqrt{}\)). Simplifying these expressions often involves techniques like rationalization and simplifying nested radicals to make them easier to work with. The goal of rationalization is to eliminate radicals from the denominator of a fraction, which is considered a standard form for mathematical expressions.

Nested radicals, like \(\sqrt{A \pm \sqrt{B}}\), can sometimes be simplified into the form \(\sqrt{x} \pm \sqrt{y}\) if \(A^2 - B\) is a perfect square. A more general method, as used in the solution, is to find \(x, y\) such that \(x+y=A\) and \(4xy=B\). This method relies on recognizing that \((\sqrt x \pm \sqrt y)^2 = x+y \pm 2\sqrt{xy}\).

Ratios are a way of comparing two quantities. A ratio \(a:b\) can also be written as a fraction \(\frac{a}{b}\). In this problem, we calculated the ratio of the product of two values (\(ab\)) to their sum (\(a+b\)), expressing it as a fraction \(\frac{ab}{a+b}\).

Solving complex problems often requires breaking them down into smaller, manageable steps. Here, simplifying the nested radical first, then the entire denominator, and finally the whole fraction allowed us to arrive at the required form \(a + \sqrt 5 b\) and determine the values of \(a\) and \(b\) accurately before calculating the final ratio.