Which expression is equivalent to $\sqrt{\frac{25 x^9 y^3}{64 x^6 y^{11}}}$ ? Assume $x>0$ and $y>0$.
A. $\frac{8 y^4 \sqrt{x}}{5 x}$
B. $\frac{8 y^2 \sqrt{x}}{5}$
C. $\frac{5 \sqrt{x}}{8 y^2}$
D. $\frac{5 x \sqrt{x}}{8 y^4}$
Answer (1)
Simplify the expression inside the square root using exponent rules: x 6 x 9 = x 3 and y 11 y 3 = y − 8 .
Rewrite the expression as 64 25 x 3 y − 8 .
Take the square root of each term: 64 25 = 8 5 , x 3 = x x , and y − 8 = y 4 1 .
Combine the simplified terms to get the final expression: 8 y 4 5 x x .
Explanation
Understanding the problem We are asked to find an expression equivalent to 64 x 6 y 11 25 x 9 y 3 , assuming that 0"> x > 0 and 0"> y > 0 . We will simplify the expression inside the square root first, and then take the square root.
Simplifying the fraction First, we simplify the fraction inside the square root by using the quotient rule for exponents, which states that x b x a = x a − b . Applying this rule to the x terms, we have x 6 x 9 = x 9 − 6 = x 3 . Applying this rule to the y terms, we have y 11 y 3 = y 3 − 11 = y − 8 . Thus, we have 64 x 6 y 11 25 x 9 y 3 = 64 25 x 3 y − 8
Taking the square root Now we can rewrite the expression as 64 25 x 3 y − 8 = 64 25 ⋅ x 3 ⋅ y − 8 Since 64 25 = 8 5 , x 3 = x 3/2 = x 1 + 1/2 = x ⋅ x 1/2 = x x , and y − 8 = y − 8/2 = y − 4 = y 4 1 , we have 64 25 x 3 y − 8 = 8 5 ⋅ x x ⋅ y 4 1 = 8 y 4 5 x x
Final Answer Therefore, the expression equivalent to 64 x 6 y 11 25 x 9 y 3 is 8 y 4 5 x x .
Examples
When calculating the flow rate of a fluid through a pipe, you might encounter expressions involving square roots and exponents similar to the one in this problem. Simplifying such expressions allows engineers to more easily determine the optimal pipe size and flow conditions, ensuring efficient and safe operation of fluid systems. This type of simplification is also useful in physics when dealing with quantities like kinetic energy or gravitational potential energy, where variables are often raised to different powers and combined in complex ways.
