Rationalize the denominator and simplify.
$\sqrt[4]{\frac{3 a^{13}}{2 b}}$
Assume that all variables represent positive numbers.
Answer (2)
Rewrite the expression as a quotient of fourth roots: 4 2 b 3 a 13 = 4 2 b 4 3 a 13 .
Rationalize the denominator by multiplying the numerator and denominator by 4 ( 2 b ) 3 : 4 2 b 4 3 a 13 ⋅ 4 ( 2 b ) 3 4 ( 2 b ) 3 = 2 b 4 3 a 13 ( 2 b ) 3 .
Simplify the numerator: 4 24 a 13 b 3 = a 3 4 24 a b 3 .
The final simplified expression is: 2 b a 3 4 24 a b 3 .
Explanation
Understanding the Problem We are given the expression 4 2 b 3 a 13 and we want to rationalize the denominator and simplify it. We assume that all variables represent positive numbers.
Rewriting the Expression First, we rewrite the expression as a quotient of fourth roots: 4 2 b 3 a 13 = 4 2 b 4 3 a 13 .
Rationalizing the Denominator To rationalize the denominator, we need to multiply both the numerator and the denominator by a factor that will make the denominator a perfect fourth power. In this case, we need to multiply by 4 ( 2 b ) 3 = 4 8 b 3 since 4 2 b ⋅ 4 ( 2 b ) 3 = 4 ( 2 b ) 4 = 2 b . So we have: 4 2 b 4 3 a 13 ⋅ 4 ( 2 b ) 3 4 ( 2 b ) 3 = 4 ( 2 b ) 4 4 3 a 13 ( 2 b ) 3 = 2 b 4 3 a 13 ( 8 b 3 ) .
Simplifying the Numerator Now we simplify the numerator. We have a 13 = a 12 ⋅ a = ( a 3 ) 4 ⋅ a . Thus, 4 3 a 13 ( 8 b 3 ) = 4 24 a 13 b 3 = 4 24 a 12 a b 3 = a 3 4 24 a b 3 .
Final Expression Substituting this back into the expression, we get: 2 b a 3 4 24 a b 3 .
Final Answer Therefore, the simplified expression with a rationalized denominator is 2 b a 3 4 24 a b 3 .
Examples
Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. This is particularly useful in fields like physics and engineering where simplified expressions are easier to work with. For example, when calculating impedance in electrical circuits or when dealing with wave functions in quantum mechanics, rationalizing denominators can lead to more manageable and interpretable results.
To rationalize the denominator of 4 2 b 3 a 13 , we rewrite it as a quotient of fourth roots, rationalize the denominator by multiplying by 4 ( 2 b ) 3 , and simplify. The final expression is 2 b a 3 4 24 a b 3 .
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