Problem 54
Question
Rationalize the denominator. $$ \frac{11}{\sqrt{7}-\sqrt{3}} $$
Step-by-Step Solution
Verified Answer
The rationalized form of this fraction is \((11\sqrt{7} + 11\sqrt{3})/4\).
1Step 1: Determine the Conjugate
The conjugate of \(\sqrt{7}-\sqrt{3}\) is \(\sqrt{7}+\sqrt{3}\). The conjugate is formed by changing the sign between the two terms.
2Step 2: Multiply by the Conjugate
Multiply both the numerator and denominator of the fraction by the conjugate. Remember that what we multiply in the denominator should also be multiplied in the numerator. So, it becomes \((11 \cdot (\sqrt{7}+\sqrt{3})) / ((\sqrt{7}-\sqrt{3}) \cdot (\sqrt{7}+\sqrt{3}))\)
3Step 3: Simplify
Carry out the multiplication in the numerator. In the denominator, use the formula \( (a+b)(a-b) = a^2 - b^2 \). So, the denominator becomes \( (\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4\). This gives us \((11\sqrt{7} + 11\sqrt{3})/4\).
Key Concepts
Conjugate MultiplicationSimplifying RadicalsDifference of Squares
Conjugate Multiplication
When rationalizing the denominator of a fraction that involves a square root, conjugate multiplication comes as a handy tool. The conjugate of an expression is formed by changing the sign between the two terms. For example, the conjugate of \(\sqrt{7} - \sqrt{3}\) is \(\sqrt{7} + \sqrt{3}\). This process is carried out to remove the radicals (square roots) from the denominator.
Conjugate multiplication involves multiplying both the numerator and the denominator of the fraction by the conjugate of the denominator. In this exercise, the fraction \(\frac{11}{\sqrt{7} - \sqrt{3}}\) must be multiplied by \(\frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}\). This is equivalent to multiplying the entire fraction by 1, which does not change its value.
This step is crucial as it sets the foundation for simplifying the denominator and reaching a rational form.
Conjugate multiplication involves multiplying both the numerator and the denominator of the fraction by the conjugate of the denominator. In this exercise, the fraction \(\frac{11}{\sqrt{7} - \sqrt{3}}\) must be multiplied by \(\frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}\). This is equivalent to multiplying the entire fraction by 1, which does not change its value.
This step is crucial as it sets the foundation for simplifying the denominator and reaching a rational form.
Simplifying Radicals
Radicals, or square roots, can appear complex, but understanding their simplification is essential. To simplify radicals, you break them down into their prime factors and look for perfect squares.
In this exercise, while the numerator \(11(\sqrt{7} + \sqrt{3})\) can appear tricky, each term inside the bracket can be handled separately by common distribution. Thus, the expression becomes \(11\sqrt{7} + 11\sqrt{3}\).
It's easiest to work with radicals once they're in a common form, allowing you to simplify expressions by combining like terms when possible. In our case, multiplication has already been done, so the expression remains as it is above. The denominator, on the other hand, simplifies through a different method called the difference of squares.
In this exercise, while the numerator \(11(\sqrt{7} + \sqrt{3})\) can appear tricky, each term inside the bracket can be handled separately by common distribution. Thus, the expression becomes \(11\sqrt{7} + 11\sqrt{3}\).
It's easiest to work with radicals once they're in a common form, allowing you to simplify expressions by combining like terms when possible. In our case, multiplication has already been done, so the expression remains as it is above. The denominator, on the other hand, simplifies through a different method called the difference of squares.
Difference of Squares
When multiplying the conjugate in the denominator, a useful algebraic identity comes into play - the difference of squares. The formula \((a + b)(a - b) = a^2 - b^2\) simplifies expressions effectively when dealing with conjugates.
In this exercise, the multiplication \((\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})\) in our denominator transforms by applying \(a^2 - b^2\):
By recognizing and employing the difference of squares, you can manage the rationalization process with ease and accuracy.
In this exercise, the multiplication \((\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})\) in our denominator transforms by applying \(a^2 - b^2\):
- \((\sqrt{7})^2 = 7\)
- \((\sqrt{3})^2 = 3\)
- Therefore, \(7 - 3 = 4\)
By recognizing and employing the difference of squares, you can manage the rationalization process with ease and accuracy.
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