The expression \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\) is equal to:
5(3 + 2√2)
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.
| Step | Process | Expression |
|---|---|---|
| 1 | Original Expression | \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\) |
| 2 | Simplify Denominator Terms | \(\sqrt{10} + 2\sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4\sqrt{5}\) |
| 3 | Combine Denominator Terms | \(3\sqrt{10} - 3\sqrt{5} = 3(\sqrt{10} - \sqrt{5})\) |
| 4 | Rewrite Expression | \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{3\left( {\sqrt {10} - \sqrt 5 } \right)}} = \frac{{5\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\left( {\sqrt {10} - \sqrt 5 } \right)}}\) |
| 5 | Rationalize Denominator | \(\frac{{5\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\left( {\sqrt {10} - \sqrt 5 } \right)}} \times \frac{{\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\left( {\sqrt {10} + \sqrt 5 } \right)}}\) |
| 6 | Simplify Numerator | \(5(\sqrt{10} + \sqrt{5})^2 = 5(10 + 5 + 2\sqrt{50}) = 5(15 + 10\sqrt{2}) = 25(3 + 2\sqrt{2})\) |
| 7 | Simplify Denominator | \((\sqrt{10} - \sqrt{5})(\sqrt{10} + \sqrt{5}) = 10 - 5 = 5\) |
| 8 | Final Result | \(\frac{{25(3 + 2\sqrt{2})}}{5} = 5(3 + 2\sqrt{2})\) |
Revision Table: Key Radical Simplification Concepts
| Concept | Description | Example |
|---|---|---|
| Simplifying Radicals | Write the number under the square root as a product of a perfect square and another number. Take the square root of the perfect square. | \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\) |
| Combining Like Radicals | Add or subtract terms that have the same radical part (same number under the square root). | \(3\sqrt{5} + 2\sqrt{5} = (3+2)\sqrt{5} = 5\sqrt{5}\) |
| Multiplying Radicals | \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\) and \(c\sqrt{a} \times d\sqrt{b} = cd\sqrt{ab}\) | \(\sqrt{3} \times \sqrt{7} = \sqrt{21}\); \(2\sqrt{3} \times 4\sqrt{5} = 8\sqrt{15}\) |
| Conjugate of a Binomial with Radicals | For a binomial \(a + \sqrt{b}\) or \(a + c\sqrt{d}\), the conjugate is \(a - \sqrt{b}\) or \(a - c\sqrt{d}\). Used for rationalizing denominators. | The conjugate of \(2 + \sqrt{3}\) is \(2 - \sqrt{3}\). |
| Rationalizing Denominators | Multiplying the numerator and denominator by a factor (often the conjugate) to remove radicals from the denominator. | \(\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\); \(\frac{1}{2+\sqrt{3}} = \frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} = \frac{2-\sqrt{3}}{4-3} = 2-\sqrt{3}\) |
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.
Important Questions from Surds and Indices
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
Let \(x = \left( {\frac{{√ {1875} }}{{√ {3888} }} \div \frac{{√ {1200} }}{{\sqrt 768}}} \right) \times \frac{{√ {175} }}{{√ {1792} }}\) . Then √x is equal to:
If \(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\) where x > 0, then the value of x is equal to:
What is the value of \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
Which of the following given value is greater than \(\sqrt[3]{12} \) ?
