Rationalize the denominator. Simplify.
$\frac{3}{15+\sqrt{10}}=\square$
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator: 15 + 10 3 × 15 − 10 15 − 10 .
Expand the denominator using the difference of squares: ( 15 + 10 ) ( 15 − 10 ) = 215 .
Distribute the 3 in the numerator: 3 ( 15 − 10 ) = 45 − 3 10 .
Simplify the expression: 215 45 − 3 10 .
Explanation
Understanding the Problem We are given the expression 15 + 10 3 and our goal is to rationalize the denominator and simplify the expression. Rationalizing the denominator means eliminating the square root from the denominator.
Multiplying by the Conjugate To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 15 + 10 is 15 − 10 . So, we multiply the given expression by 15 − 10 15 − 10 : 15 + 10 3 × 15 − 10 15 − 10 = ( 15 + 10 ) ( 15 − 10 ) 3 ( 15 − 10 )
Expanding the Denominator Now, we expand the denominator using the difference of squares formula, which states that ( a + b ) ( a − b ) = a 2 − b 2 . In our case, a = 15 and b = 10 . So, the denominator becomes: 1 5 2 − ( 10 ) 2 = 225 − 10 = 215 Thus, our expression becomes: 215 3 ( 15 − 10 )
Distributing in the Numerator Next, we distribute the 3 in the numerator: 215 3 × 15 − 3 × 10 = 215 45 − 3 10
Simplifying the Expression Finally, we check if the fraction can be simplified further. We look for common factors between 45, 3, and 215. The prime factorization of 45 is 3 2 × 5 , of 3 is 3, and of 215 is 5 × 43 . Since there are no common factors other than 1, the expression is in its simplest form. Therefore, the rationalized and simplified expression is: 215 45 − 3 10
Final Answer The rationalized and simplified form of the given expression is 215 45 − 3 10 .
Examples
Rationalizing the denominator is a useful technique in various mathematical and scientific contexts. For instance, when calculating impedance in electrical circuits or dealing with complex numbers in quantum mechanics, it's often necessary to remove square roots from the denominator to simplify calculations and make results easier to interpret. This technique ensures that the expression is in a standard form, which is essential for further analysis and comparison with other results. By rationalizing the denominator, we make the expression easier to work with and understand, facilitating further calculations and interpretations.
