Simplify each expression:
a) $\frac{17}{\sqrt{10}}=$\square
b) $\sqrt{\frac{343}{20}}=$\square
c) $\frac{15}{\sqrt{5 c}}=$\square
Note: To input $a \sqrt{b}$ type "a sqrt(b)". To input $a \sqrt[b]{c}$ type "a root(b)(c)".
Answer (1)
Rationalize the denominator of 10 17 by multiplying the numerator and denominator by 10 , resulting in 10 17 10 .
Simplify 20 343 by factoring and rationalizing the denominator, resulting in 10 7 35 .
Rationalize the denominator of 5 c 15 by multiplying the numerator and denominator by 5 c , resulting in c 3 5 c .
The simplified expressions are 10 17 10 , 10 7 35 , and c 3 5 c .
Explanation
Understanding the Problem We are given three expressions to simplify and rationalize their denominators.
Simplifying the first expression a) To simplify 10 17 , we multiply both the numerator and the denominator by 10 to rationalize the denominator: 10 17 = 10 × 10 17 × 10 = 10 17 10
Simplifying the second expression b) To simplify 20 343 , we first simplify the fraction inside the square root. We have 343 = 7 3 = 49 × 7 and 20 = 4 × 5 . Thus, 20 343 = 4 × 5 49 × 7 = 4 × 5 49 × 7 = 2 5 7 7 Now, we rationalize the denominator by multiplying both the numerator and the denominator by 5 : 2 5 7 7 = 2 5 × 5 7 7 × 5 = 2 × 5 7 35 = 10 7 35
Simplifying the third expression c) To simplify 5 c 15 , we multiply both the numerator and the denominator by 5 c to rationalize the denominator: 5 c 15 = 5 c × 5 c 15 × 5 c = 5 c 15 5 c = c 3 5 c
Final Answer Therefore, the simplified expressions are: a) 10 17 10 b) 10 7 35 c) c 3 5 c
Examples
Rationalizing denominators is a useful skill in various fields, such as physics and engineering, where simplified expressions can make calculations easier. For example, when calculating the impedance of an electrical circuit, you might encounter expressions with square roots in the denominator. Rationalizing the denominator can help in simplifying the expression and making it easier to work with. Similarly, in mechanics, when dealing with forces or velocities, rationalizing denominators can lead to more manageable equations.
