complex square root 14 over complex square root 77 symbols wont come out but 14 is over 77 fraction
The Correct Answer and Explanation is :
To compute the given expression:
[
\frac{\sqrt{14}}{\sqrt{77}}
]
Step-by-step Solution:
- Simplify the square root of the fraction:
By the property of square roots, (\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}):
[
\frac{\sqrt{14}}{\sqrt{77}} = \sqrt{\frac{14}{77}}
] - Simplify the fraction inside the square root:
Divide the numerator and denominator by their greatest common divisor (GCD), which is 7:
[
\frac{14}{77} = \frac{2}{11}
] So, the expression becomes:
[
\sqrt{\frac{2}{11}}
] - Express the square root of the fraction:
Split the square root:
[
\sqrt{\frac{2}{11}} = \frac{\sqrt{2}}{\sqrt{11}}
] - Rationalize the denominator:
To remove the square root from the denominator, multiply numerator and denominator by (\sqrt{11}):
[
\frac{\sqrt{2}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{\sqrt{22}}{11}
]
Final Answer:
[
\frac{\sqrt{14}}{\sqrt{77}} = \frac{\sqrt{22}}{11}
]
Explanation (300 Words):
The problem involves simplifying a ratio of two square roots. The process follows mathematical conventions to ensure the result is in its simplest and most acceptable form.
First, the property of square roots, (\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}), simplifies the expression into a single square root. This is a standard approach to combine terms under one root, making further simplification easier.
Next, the fraction inside the square root, (\frac{14}{77}), is reduced by dividing both numerator and denominator by their greatest common divisor, which is 7. Simplifying fractions is an essential step, as it reduces complexity and avoids unnecessarily large numbers.
After simplifying to (\sqrt{\frac{2}{11}}), the square root is split again for clarity as (\frac{\sqrt{2}}{\sqrt{11}}). At this point, the denominator contains a square root, which is traditionally rationalized. Rationalization is performed to make the denominator a rational number, ensuring the result adheres to conventional mathematical standards.
Multiplying numerator and denominator by (\sqrt{11}) achieves this, yielding (\frac{\sqrt{22}}{11}). This process does not change the value of the expression but transforms it into a more standard form.
The final result, (\frac{\sqrt{22}}{11}), is the simplest representation. Rationalizing denominators is common practice in mathematics to improve clarity and alignment with conventions, especially in higher-level work where clear communication of results is vital.