Simplify Radical Expression Step-by-Step

We are asked to simplify the given mathematical expression involving square roots:

\(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\)

Analyzing the Radical Expression

The expression has a numerator \(15\left( {\sqrt {10} + \sqrt 5 } \right)\) and a denominator \(\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80 }\). The denominator contains several terms with square roots that can be simplified.

Simplifying Terms in the Denominator

Let's simplify each term in the denominator where possible:

• \(\sqrt{10}\) is already in its simplest form.
• \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\)
• \(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\)
• \(\sqrt{5}\) is already in its simplest form.
• \(\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\)

Combining Like Terms in the Denominator

Now, substitute the simplified terms back into the denominator:

\(\sqrt {10} + 2\sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4\sqrt{5}\)

Group the terms with \(\sqrt{10}\) and \(\sqrt{5}\):

\((\sqrt{10} + 2\sqrt{10}) + (2\sqrt{5} - \sqrt{5} - 4\sqrt{5})\)

Combine the coefficients of the like terms:

\((1+2)\sqrt{10} + (2-1-4)\sqrt{5}\)

\(3\sqrt{10} - 3\sqrt{5}\)

We can factor out 3 from the denominator:

\(3(\sqrt{10} - \sqrt{5})\)

Rewrite the Expression

Substitute the simplified denominator back into the original expression:

\(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{3\left( {\sqrt {10} - \sqrt 5 } \right)}}\)

We can cancel the common factor of 3 from the numerator (15) and the denominator (3):

\(\frac{{5\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\left( {\sqrt {10} - \sqrt 5 } \right)}}\)

Rationalize the Denominator

To eliminate the radical from the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The denominator is \(\sqrt{10} - \sqrt{5}\), so its conjugate is \(\sqrt{10} + \sqrt{5}\).

\(\frac{{5\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\left( {\sqrt {10} - \sqrt 5 } \right)}} \times \frac{{\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\left( {\sqrt {10} + \sqrt 5 } \right)}}\)

Expand and Simplify Numerator

Numerator: \(5 \times (\sqrt{10} + \sqrt{5})^2\)

Using the formula \((a+b)^2 = a^2 + 2ab + b^2\):

\(5 \times ((\sqrt{10})^2 + 2(\sqrt{10})(\sqrt{5}) + (\sqrt{5})^2)\)

\(5 \times (10 + 2\sqrt{50} + 5)\)

\(5 \times (15 + 2\sqrt{25 \times 2})\)

\(5 \times (15 + 2 \times 5\sqrt{2})\)

\(5 \times (15 + 10\sqrt{2})\)

Distribute the 5:

\(75 + 50\sqrt{2}\)

Alternatively, keep 5 factored out: \(5(15 + 10\sqrt{2}) = 5 \times 5(3 + 2\sqrt{2}) = 25(3 + 2\sqrt{2})\)

Expand and Simplify Denominator

Denominator: \((\sqrt{10} - \sqrt{5})(\sqrt{10} + \sqrt{5})\)

Using the formula \((a-b)(a+b) = a^2 - b^2\):

\((\sqrt{10})^2 - (\sqrt{5})^2\)

\(10 - 5 = 5\)

Final Simplification

Now, put the simplified numerator over the simplified denominator:

\(\frac{{25(3 + 2\sqrt{2})}}{5}\)

Cancel the common factor of 5:

\(5(3 + 2\sqrt{2})\)

Comparing this result with the given options, we find that it matches the first option.

Revision Table: Key Radical Simplification Concepts

Additional Information: Understanding Surds and Radical Expressions

A surd is a root of a number that cannot be expressed as a simple fraction. For example, \(\sqrt{2}\) and \(\sqrt{5}\) are surds, but \(\sqrt{4}\) (which equals 2) is not.

Radical expressions are expressions that contain square roots (or other roots). Simplifying radical expressions involves rewriting them in a simpler form, often by simplifying the numbers under the radical sign and by rationalizing denominators.

Rationalizing the denominator is a standard technique used to simplify expressions and make further calculations easier, especially when dealing with fractions involving radicals. It ensures that the denominator is a rational number.

When simplifying expressions with multiple radical terms, identify and combine 'like' terms. Like terms have the same radical part, such as \(3\sqrt{7}\) and \(5\sqrt{7}\).

Remember the algebraic identities, like the difference of squares \((a-b)(a+b) = a^2 - b^2\) and the perfect square formulas \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\), as they are very useful in simplifying expressions involving radicals, especially during rationalization.