Which expression is equivalent to [tex]x y^{\frac{2}{9}}[/tex] ?
A. [tex]\sqrt{x y^9}[/tex]
B. [tex]\sqrt[9]{x y^2}[/tex]
C. [tex]x\left(\sqrt{y^9}\right)[/tex]
D. [tex]x\left(\sqrt[9]{y^2}\right)[/tex]
Answer (2)
Rewrite the fractional exponent as a radical: y 9 2 = 9 y 2 .
Substitute the radical form into the original expression: x y 9 2 = x 9 y 2 .
Compare the result with the given options.
The equivalent expression is x ( 9 y 2 ) . x ( 9 y 2 )
Explanation
Understanding the Problem We are given the expression x y 9 2 and asked to find an equivalent expression from the given options. The key here is to understand how fractional exponents relate to radicals.
Rewriting the Fractional Exponent Recall that a n m can be written as n a m or ( n a ) m . In our case, we have y 9 2 , which can be rewritten as 9 y 2 or ( 9 y ) 2 .
Substituting the Radical Form Now, we can rewrite the entire expression x y 9 2 as x 9 y 2 .
Identifying the Equivalent Expression Comparing this with the given options:
x y 9 is not equivalent.
9 x y 2 is not equivalent.
x ( y 9 ) is not equivalent.
x ( 9 y 2 ) is equivalent.
Final Answer Therefore, the expression equivalent to x y 9 2 is x ( 9 y 2 ) .
Examples
Understanding fractional exponents and radicals is crucial in various fields, such as physics and engineering. For instance, when calculating the period of a pendulum, the formula involves a square root, which is a fractional exponent. Similarly, in electrical engineering, impedance calculations often involve complex numbers and fractional exponents. By mastering these concepts, students can tackle real-world problems involving oscillations, waves, and electrical circuits more effectively.
The expression equivalent to x y 9 2 is x ( 9 y 2 ) , which aligns with option D. This is found by rewriting the fractional exponent as a radical form. The other options do not match this equivalent expression.
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