Simplify each expression:
a) $\frac{5}{\sqrt{6}}=$ $\square$
b) $\sqrt{\frac{17}{875}}=$ $\square$
c) $\frac{102}{\sqrt{17 m}}=$ $\square$

Rationalize the denominator of 6 ​ 5 ​ by multiplying the numerator and denominator by 6 ​ , resulting in 6 5 6 ​ ​ .
Simplify 875 17 ​ ​ by simplifying the fraction inside the square root and rationalizing the denominator, resulting in 175 595 ​ ​ .
Rationalize the denominator of 17 m ​ 102 ​ by multiplying the numerator and denominator by 17 m ​ and simplifying, resulting in m 6 17 m ​ ​ .
The simplified expressions are 6 5 6 ​ ​ ​ , 175 595 ​ ​ ​ , and m 6 17 m ​ ​ ​ .

Explanation

Problem Analysis We are asked to simplify three expressions involving square roots. We will rationalize denominators and simplify fractions to achieve this.

Simplifying Expression a a) To simplify 6 ​ 5 ​ , we rationalize the denominator by multiplying both the numerator and denominator by 6 ​ : 6 ​ 5 ​ = 6 ​ × 6 ​ 5 × 6 ​ ​ = 6 5 6 ​ ​

Simplifying Expression b b) To simplify 875 17 ​ ​ , we first simplify the fraction inside the square root. The prime factorization of 875 is 5 3 × 7 = 25 × 35 . Thus, we have: 875 17 ​ ​ = 5 3 × 7 17 ​ ​ = 25 × 35 17 ​ ​ = 25 × 35 ​ 17 ​ ​ = 5 35 ​ 17 ​ ​ Now, we rationalize the denominator by multiplying both the numerator and the denominator by 35 ​ : 5 35 ​ 17 ​ ​ = 5 35 ​ × 35 ​ 17 ​ × 35 ​ ​ = 5 × 35 17 × 35 ​ ​ = 175 595 ​ ​

Simplifying Expression c c) To simplify 17 m ​ 102 ​ , we rationalize the denominator by multiplying both the numerator and the denominator by 17 m ​ : 17 m ​ 102 ​ = 17 m ​ × 17 m ​ 102 × 17 m ​ ​ = 17 m 102 17 m ​ ​ Since 102 = 6 × 17 , we can simplify the fraction: 17 m 102 17 m ​ ​ = 17 m 6 × 17 17 m ​ ​ = m 6 17 m ​ ​

Final Answer Therefore, the simplified expressions are: a) 6 5 6 ​ ​ b) 175 595 ​ ​ c) m 6 17 m ​ ​


Examples
Simplifying radical expressions is a fundamental skill in algebra and is used in various fields such as physics, engineering, and computer graphics. For example, when calculating distances or areas involving irrational numbers, simplifying radical expressions helps in obtaining more manageable and accurate results. In physics, simplifying expressions with square roots can arise when dealing with energy calculations or wave phenomena. In computer graphics, these simplifications can optimize calculations for rendering images and animations.

Answered by GinnyAnswer | 2025-07-07

The simplified expressions are 6 5 √ 6 ​ , 175 √ 595 ​ , and m 6 √ ( 17 m ) ​ . Each expression was simplified by rationalizing the denominator and simplifying fractions where necessary.
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Answered by Anonymous | 2025-07-28