Rationalize the denominator. Simplify.
$\frac{17}{13-\sqrt{10}}=$
$\square$
NOTE: To input $a \sqrt{b}$, type "a sqrt(b)".
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator: 13 − 10 17 ⋅ 13 + 10 13 + 10 .
Expand the numerator: 17 ( 13 + 10 ) = 221 + 17 10 .
Expand the denominator: ( 13 − 10 ) ( 13 + 10 ) = 169 − 10 = 159 .
The simplified expression is: 159 221 + 17 10 .
Explanation
Understanding the Problem We are given the expression 13 − 10 17 and our goal is to rationalize the denominator and simplify the expression. Rationalizing the denominator means we want to eliminate 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 13 − 10 is 13 + 10 . So, we multiply the given expression by 13 + 10 13 + 10 : 13 − 10 17 ⋅ 13 + 10 13 + 10 .
Expanding the Numerator Now, we multiply out the numerator: 17 ( 13 + 10 ) = 17 ⋅ 13 + 17 10 = 221 + 17 10 .
Expanding the Denominator Next, we multiply out the denominator: ( 13 − 10 ) ( 13 + 10 ) = 1 3 2 − ( 10 ) 2 = 169 − 10 = 159 .
Combining the Results So, the expression becomes: 159 221 + 17 10 .
Checking for Simplification Now, we check if the fraction can be simplified. The prime factorization of 159 is 3 × 53 . Since 17 is a prime number and 17 does not divide 159 or 221, the fraction is already in its simplest form. Therefore, the rationalized and simplified expression is: 159 221 + 17 10 .
Final Answer Therefore, the final answer is 159 221 + 17 10 .
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, where simplified expressions are crucial for calculations. For example, when dealing with impedance in electrical circuits or calculating forces in mechanics, having a rationalized denominator can make subsequent calculations easier and more accurate. This skill is also fundamental in more advanced mathematical studies, such as calculus and complex analysis, where simplifying expressions is a common task.
