Rewrite with rational exponents.
$\left(\sqrt[9]{2 x^4 y}\right)^{11}$
Answer (1)
Rewrite the expression inside the parenthesis using a rational exponent: 9 2 x 4 y = ( 2 x 4 y ) 9 1 .
Substitute back into the original expression: ( ( 2 x 4 y ) 9 1 ) 11 .
Apply the power of a power rule: ( 2 x 4 y ) 9 1 ⋅ 11 = ( 2 x 4 y ) 9 11 .
Distribute the exponent: ( 2 x 4 y ) 9 11 = 2 9 11 x 9 44 y 9 11 . The final answer is 2 9 11 x 9 44 y 9 11 .
Explanation
Understanding the Problem We are asked to rewrite the expression ( 9 2 x 4 y ) 11 using rational exponents. This involves understanding fractional exponents and applying the power of a power rule.
Converting to Rational Exponent First, we rewrite the expression inside the parenthesis using a rational exponent. Recall that n a = a n 1 . Therefore, 9 2 x 4 y = ( 2 x 4 y ) 9 1 .
Substituting Back Now, substitute this back into the original expression: ( ( 2 x 4 y ) 9 1 ) 11 .
Applying Power of a Power Rule Next, we use the power of a power rule, which states that ( a m ) n = a m ⋅ n . Applying this rule, we get: ( 2 x 4 y ) 9 1 ⋅ 11 = ( 2 x 4 y ) 9 11 .
Distributing the Exponent Finally, we distribute the exponent to each term inside the parenthesis: ( 2 x 4 y ) 9 11 = 2 9 11 ( x 4 ) 9 11 y 9 11 = 2 9 11 x 9 4 ⋅ 11 y 9 11 = 2 9 11 x 9 44 y 9 11 .
Final Answer Therefore, the expression ( 9 2 x 4 y ) 11 rewritten with rational exponents is 2 9 11 x 9 44 y 9 11 .
Examples
Rational exponents are useful in various fields, such as physics and engineering, when dealing with equations involving roots and powers. For example, when calculating the period of a pendulum, the formula involves a square root, which can be expressed as a rational exponent. Similarly, in electrical engineering, when analyzing circuits with inductors and capacitors, the impedance calculations often involve square roots and other radicals, making rational exponents a valuable tool for simplifying and manipulating these expressions. Understanding how to manipulate expressions with rational exponents allows engineers and physicists to simplify complex equations and solve problems more efficiently.
