Rewrite without rational exponents, and simplify, if possible.
[tex]$\left(a^2 b^2\right)^{\frac{1}{9}}=\square$[/tex]
(Simplify your answer. Type in radical form)
Answer (2)
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 asked to rewrite the expression ( a 2 b 2 ) 9 1 without rational exponents and simplify it.
Plan of attack We will use the properties of exponents and radicals to rewrite the given expression. Specifically, we'll use the power of a product rule and the power of a power rule.
Applying exponent and radical rules First, we apply the power of a product rule: ( x y ) n = x n y n . This gives us ( a 2 b 2 ) 9 1 = ( a 2 ) 9 1 ( b 2 ) 9 1 Next, we apply the power of a power rule: ( x m ) n = x mn . This simplifies the exponents: ( a 2 ) 9 1 ( b 2 ) 9 1 = a 9 2 b 9 2 Now, we rewrite the expression using radicals. Recall that x n m = n x m . Thus, a 9 2 b 9 2 = 9 a 2 9 b 2 Finally, we 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 Therefore, the simplified expression is 9 a 2 b 2 .
Examples
Understanding how to simplify expressions with rational exponents and radicals is useful in various fields, such as physics and engineering, when dealing with complex formulas. For example, when calculating the area or volume of objects with non-standard shapes, you might encounter expressions involving radicals and rational exponents. Simplifying these expressions makes the calculations easier and more manageable. Also, in computer graphics, transformations involving scaling and rotations often use exponents and radicals, and simplifying these expressions can improve the efficiency of the rendering process.
To simplify ( a 2 b 2 ) 9 1 , we use the power of a product and power of a power rules, leading to 9 a 2 b 2 . Thus, the expression can be rewritten without rational exponents as 9 a 2 b 2 .
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