Final answer:
The expression √(75 m⁹)/(196 m⁴) simplifies to (5 m²)/(14√m), which is achieved by factoring the numerical parts, simplifying the variable parts using properties of exponents, and converting the result back to radical form.
Step-by-step explanation:
To simplify the expression √(75 m⁹)/(196 m⁴), start by factoring the numerical parts. The number 75 can be factored into 25 × 3, and 25 is a perfect square. The number 196 is also a perfect square, being 14 × 14. Next, consider the variable part, m, which appears as m⁹ in the numerator and m⁴ in the denominator. We can use the properties of exponents to simplify this by subtracting the powers (since we're dividing like bases).
The expression then becomes:
- √(25 × 3 × m⁹) / (14 × 14 × m⁴)
- = (√25) √3 √(m⁹) / (14 m⁴)
- = 5√3 m 9/2 / 14 m⁴
- We must simplify further by subtracting the exponents of m: (9/2) - 4 = (9/2) - (8/2) = 1/2.
- The simplified form of the expression is: (5√3 m 1/2) / 14
Converting m 1/2 back to its radical form, we get √m. So the final expression is:
(5 m²) / (14√m)
Option a, (5 m²)/(14√m), is the correct simplification.