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. $\frac{5 \sqrt{x}}{8 y^2}$
Answer (2)
Simplify the fraction inside the square root using exponent rules: 64 x 6 y 11 25 x 9 y 3 = 64 y 8 25 x 3 .
Take the square root of the simplified fraction: 64 y 8 25 x 3 = 8 y 4 5 x 3/2 .
Rewrite the expression using x 3/2 = x x : 8 y 4 5 x x .
Compare the simplified expression with the given options. None of the options are equivalent to the simplified expression. The correct simplification is 8 y 4 5 x x .
Explanation
Understanding the Problem We are given the expression 64 x 6 y 11 25 x 9 y 3 and the conditions 0"> x > 0 and 0"> y > 0 . We want to find an equivalent expression among the given options.
Simplifying the Fraction First, let's simplify the fraction inside the square root. We have 64 x 6 y 11 25 x 9 y 3 . We can simplify the x and y terms using the rule x b x a = x a − b . So, we get: x 6 x 9 = x 9 − 6 = x 3 y 11 y 3 = y 3 − 11 = y − 8 = y 8 1 Thus, the fraction becomes: 64 x 6 y 11 25 x 9 y 3 = 64 y 8 25 x 3
Taking the Square Root Now, we take the square root of the simplified fraction. Recall that b a = b a and x a = x 2 a . So, we have: 64 y 8 25 x 3 = 64 y 8 25 x 3 = 64 ⋅ y 8 25 ⋅ x 3 = 8 ⋅ y 2 8 5 ⋅ x 2 3 = 8 y 4 5 x 2 3 Since x 2 3 = x 1 + 2 1 = x 1 ⋅ x 2 1 = x x , we can rewrite the expression as: 8 y 4 5 x x
Comparing with Options Now, let's compare our simplified expression 8 y 4 5 x x with the given options: Option 1: 5 x 8 y 4 x Option 2: 5 8 y 2 x Option 3: 8 y 2 5 x None of the options match our simplified expression. However, let's manipulate our expression to see if we can get it into a form that matches one of the options. We can rewrite x as x ⋅ x , so our expression becomes: 8 y 4 5 x x = 8 y 4 5 x x x This still doesn't match any of the options. It seems there might be a typo in the problem or the options.
Final Answer The closest option to our simplified expression 8 y 4 5 x x is 8 y 2 5 x . However, they are not equivalent. The correct simplification of the given expression is $\frac{5x
\sqrt{x}}{8y^4}$
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, when dealing with equations involving square roots. For example, when calculating the impedance of an electrical circuit, you might encounter a complex expression with square roots that needs to be simplified to find the overall resistance. Similarly, in physics, when dealing with projectile motion, simplifying radical expressions can help determine the range or maximum height of a projectile. These simplifications make calculations easier and provide a clearer understanding of the underlying relationships.
The expression 64 x 6 y 11 25 x 9 y 3 simplifies to 8 y 4 5 x x , which does not match any of the provided options. Thus, none of the options are equivalent to the correct simplification. The correct simplified form is not listed among the provided options.
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