Simplify using the quotient rule. Assume that all variables are positive.
[tex]$\sqrt[4]{\frac{4 y^6}{x^{12}}}$[/tex]
Answer (2)
Apply the quotient rule for radicals to separate the numerator and denominator: 4 x 12 4 y 6 = 4 x 12 4 4 y 6 .
Simplify the numerator: 4 4 y 6 = y 2 y .
Simplify the denominator: 4 x 12 = x 3 .
Combine the simplified terms to get the final answer: x 3 y 2 y .
Explanation
Understanding the problem We are asked to simplify the expression 4 x 12 4 y 6 using the quotient rule, assuming all variables are positive.
Applying the Quotient Rule The quotient rule for radicals states that n b a = n b n a . Applying this rule, we get: 4 x 12 4 y 6 = 4 x 12 4 4 y 6
Simplifying Numerator and Denominator Now we simplify the numerator and the denominator separately. For the numerator, we have: 4 4 y 6 = 4 2 2 y 6 = ( 2 2 y 6 ) 4 1 = 2 4 2 y 4 6 = 2 2 1 y 2 3 = 2 y y = y 2 y For the denominator, we have: 4 x 12 = ( x 12 ) 4 1 = x 4 12 = x 3
Combining Simplified Terms Combining the simplified numerator and denominator, we get: 4 x 12 4 4 y 6 = x 3 y 2 y
Final Answer Thus, the simplified expression is x 3 y 2 y .
Examples
Radicals and quotient rules are used in various fields like physics and engineering to simplify complex formulas. For instance, when calculating the impedance in electrical circuits or dealing with wave equations, simplifying expressions involving radicals can make the calculations more manageable. Imagine you are designing a bridge and need to calculate stress distribution; simplifying radical expressions helps in getting accurate results efficiently.
To simplify the expression 4 x 12 4 y 6 using the quotient rule, we separate the numerator and denominator. After simplification, we find that the result is x 3 y 2 y .
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