Rewrite without rational exponents, and simplify, if possible.
$\left(a^2 b^2\right)^{\frac{1}{9}}$
Answer (1)
Apply the power of a product rule: ( a 2 b 2 ) 9 1 = ( a 2 ) 9 1 ( b 2 ) 9 1 .
Apply the power of a power rule: ( a 2 ) 9 1 ( b 2 ) 9 1 = a 9 2 b 9 2 .
Rewrite using radicals: a 9 2 b 9 2 = 9 a 2 9 b 2 .
Combine the radicals: 9 a 2 9 b 2 = 9 a 2 b 2 . The final answer is 9 a 2 b 2 .
Explanation
Understanding the Problem We are given the expression ( a 2 b 2 ) 9 1 . Our goal is to rewrite this expression without rational exponents and simplify it as much as possible.
Strategy We will use the properties of exponents to simplify the given expression. Specifically, we'll use the power of a product rule, which states that ( x y ) n = x n y n , and the power of a power rule, which states that ( x m ) n = x mn .
Applying Power of a Product Rule First, apply the power of a product rule to the expression: ( a 2 b 2 ) 9 1 = ( a 2 ) 9 1 ( b 2 ) 9 1
Applying Power of a Power Rule Next, apply the power of a power rule to each term: ( a 2 ) 9 1 = a 2 × 9 1 = a 9 2 ( b 2 ) 9 1 = b 2 × 9 1 = b 9 2 So we have: a 9 2 b 9 2
Rewriting with Radicals Now, rewrite the expression using radicals. Recall that x n m = n x m . Therefore, a 9 2 = 9 a 2 b 9 2 = 9 b 2 So the expression becomes: 9 a 2 9 b 2
Combining Radicals Finally, combine the radicals using the property n x n y = n x y : 9 a 2 9 b 2 = 9 a 2 b 2
Final Answer Thus, the simplified expression without rational exponents is 9 a 2 b 2 .
Examples
Understanding how to simplify expressions with rational exponents is useful in various fields, such as physics and engineering, where complex formulas often involve exponents and radicals. For example, when calculating the period of a pendulum, the formula involves square roots and fractional exponents. Simplifying these expressions allows for easier computation and a better understanding of the relationships between different variables. Also, in computer graphics, transformations like scaling and rotations often involve matrix operations with exponents, and simplifying these expressions can optimize performance.
