Rationalize the denominator.
$\frac{-12 \sqrt{5}}{\sqrt{6}-\sqrt{3}}$
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator: 6 + 3 6 + 3 .
Simplify the denominator using the difference of squares: ( 6 − 3 ) ( 6 + 3 ) = 3 .
Simplify the numerator: − 12 5 ( 6 + 3 ) = − 12 ( 30 + 15 ) .
Divide both terms in the numerator by 3 to get the final answer: − 4 30 − 4 15 .
Explanation
Understanding the Problem We are given the expression 6 − 3 − 12 5 . Our goal is to rationalize the denominator, which means we want to eliminate any square roots 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 6 − 3 is 6 + 3 . So, we multiply the given expression by 6 + 3 6 + 3 .
Rewriting the Expression The expression becomes: 6 − 3 − 12 5 ⋅ 6 + 3 6 + 3 = ( 6 − 3 ) ( 6 + 3 ) − 12 5 ( 6 + 3 ) .
Simplifying the Denominator Now, we simplify the denominator using the difference of squares formula, ( a − b ) ( a + b ) = a 2 − b 2 : ( 6 − 3 ) ( 6 + 3 ) = ( 6 ) 2 − ( 3 ) 2 = 6 − 3 = 3.
Simplifying the Numerator Next, we simplify the numerator by distributing − 12 5 : − 12 5 ( 6 + 3 ) = − 12 ( 5 ⋅ 6 + 5 ⋅ 3 ) = − 12 ( 30 + 15 ) .
Combining the Results So, the expression now becomes: 3 − 12 ( 30 + 15 ) .
Final Simplification Finally, we divide both terms in the numerator by 3: 3 − 12 ( 30 + 15 ) = − 4 ( 30 + 15 ) = − 4 30 − 4 15 . Therefore, the rationalized expression is − 4 30 − 4 15 .
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, to simplify expressions and make calculations easier. For example, when dealing with impedance in electrical circuits or wave functions in quantum mechanics, rationalizing the denominator can help in obtaining more manageable and interpretable results. This technique is also useful in simplifying complex numbers and algebraic expressions in general.
