Rewrite without rational exponents, and simplify, if possible.
$\left(a^2 b^2\right)^{\frac{1}{9}}$
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 rule: ( a 2 ) 9 1 ( b 2 ) 9 1 = a 9 2 b 9 2 .
Rewrite the expression 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 if possible. This involves applying exponent rules and converting the rational exponent to a radical.
Applying the Power of a Product Rule First, we apply the power of a product rule, which states that ( x y ) n = x n y n . Applying this rule, we get: ( a 2 b 2 ) 9 1 = ( a 2 ) 9 1 ( b 2 ) 9 1
Applying the Power Rule Next, we apply the power rule, which states that ( x m ) n = x mn . Applying this rule to both terms, we have: ( a 2 ) 9 1 ( b 2 ) 9 1 = a 2 × 9 1 b 2 × 9 1 = a 9 2 b 9 2
Rewriting with Radicals Now, we rewrite the expression using radicals. Recall that x n m = n x m . Therefore, we can rewrite a 9 2 and b 9 2 as follows: a 9 2 = 9 a 2 b 9 2 = 9 b 2 So, the entire expression becomes: a 9 2 b 9 2 = 9 a 2 9 b 2 Since the radicals have the same index, we can combine them: 9 a 2 9 b 2 = 9 a 2 b 2 Thus, the simplified expression without rational exponents is 9 a 2 b 2 .
Final Answer Therefore, the expression ( a 2 b 2 ) 9 1 rewritten without rational exponents and simplified is 9 a 2 b 2 .
Examples
Understanding and simplifying expressions with rational exponents is useful in various fields, such as physics and engineering, where complex calculations involving roots and powers are common. For example, when calculating the period of a pendulum or analyzing the behavior of waves, simplifying expressions with rational exponents can make the calculations more manageable and provide clearer insights into the underlying phenomena. This skill is also essential in computer graphics for scaling and transforming objects, ensuring that images and models are displayed correctly and efficiently.
To rewrite ( a 2 b 2 ) 9 1 without rational exponents, we first separate the bases and apply the power rule, resulting in a 9 2 b 9 2 . Then, we convert these to radicals, yielding 9 a 2 b 2 as the final expression. Thus, the answer is 9 a 2 b 2 .
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