What is the following quotient? (9 + √2) / (4 - √7)

Understand the Problem

The question is asking for the result of dividing the expression '(9 + √2)' by '(4 - √7)'. This will involve simplifying the quotient to find the final result.

Answer

$$ 4 + \sqrt{7} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9} $$

Steps to Solve

Set up the division We begin with the expression: $$ \frac{9 + \sqrt{2}}{4 - \sqrt{7}} $$
Multiply by the conjugate To simplify, multiply both the numerator and denominator by the conjugate of the denominator, (4 + \sqrt{7}): $$ \frac{(9 + \sqrt{2})(4 + \sqrt{7})}{(4 - \sqrt{7})(4 + \sqrt{7})} $$
Calculate the denominator The denominator can be computed as follows: $$ (4 - \sqrt{7})(4 + \sqrt{7}) = 4^2 - (\sqrt{7})^2 = 16 - 7 = 9 $$
Calculate the numerator Now expand the numerator: $$ (9 + \sqrt{2})(4 + \sqrt{7}) = 9 \cdot 4 + 9 \cdot \sqrt{7} + \sqrt{2} \cdot 4 + \sqrt{2} \cdot \sqrt{7} $$ This gives: $$ 36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14} $$
Combine the results Thus, the expression simplifies to: $$ \frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9} $$
Separate the terms The final step involves separating the terms: $$ \frac{36}{9} + \frac{9\sqrt{7}}{9} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9} = 4 + \sqrt{7} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9} $$

The simplified quotient is: $$ 4 + \sqrt{7} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9} $$

More Information

This problem involves simplifying a fraction involving square roots by using the technique of multiplying by the conjugate to eliminate the square root from the denominator. The final result is a combination of rational and irrational terms.

Tips

Not using the conjugate: Failing to multiply by the conjugate can lead to a more complicated expression.
Errors in expanding: Ensure that all terms in the numerator are properly multiplied when expanding the expression.

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