What is $\frac{\left(x^2 y^2\right)^{\frac{1}{2}}}{\sqrt[3]{x^2 y}}$ in exponential form?
Answer (2)
The expression 3 x 2 y ( x 2 y 2 ) 2 1 simplifies to x 3 1 y 3 2 in exponential form. This is achieved by simplifying the numerator and denominator using exponent rules and then dividing them. The final result is x 3 1 y 3 2 .
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Simplify the numerator: ( x 2 y 2 ) 2 1 = x y .
Simplify the denominator: 3 x 2 y = x 3 2 y 3 1 .
Divide the numerator by the denominator: x 3 2 y 3 1 x y = x 3 1 y 3 2 .
The simplified expression is x 3 1 y 3 2 .
Explanation
Understanding the Problem We want to simplify the expression 3 x 2 y ( x 2 y 2 ) 2 1 and express it in exponential form. This involves applying exponent rules to both the numerator and the denominator and then simplifying the resulting expression.
Simplifying the Numerator First, let's simplify the numerator: ( x 2 y 2 ) 2 1 . Using the power of a product rule, we have: ( x 2 y 2 ) 2 1 = x 2 ⋅ 2 1 y 2 ⋅ 2 1 = x 1 y 1 = x y
Simplifying the Denominator Next, let's simplify the denominator: 3 x 2 y . We can rewrite the cube root as a fractional exponent: 3 x 2 y = ( x 2 y ) 3 1 Now, apply the power of a product rule: ( x 2 y ) 3 1 = x 3 2 y 3 1
Dividing Numerator by Denominator Now, we divide the simplified numerator by the simplified denominator: x 3 2 y 3 1 x y = x 3 2 y 3 1 x 1 y 1 Using the quotient rule for exponents, we subtract the exponents of like bases: x 1 − 3 2 y 1 − 3 1 = x 3 3 − 3 2 y 3 3 − 3 1 = x 3 1 y 3 2
Finding the Correct Match The simplified expression in exponential form is x 3 1 y 3 2 . Now we compare this with the given options to find the correct match.
The correct answer is x 3 1 y 3 2 .
Examples
Understanding exponential forms is crucial in various fields like physics and engineering. For instance, when dealing with wave functions in quantum mechanics, you often encounter expressions involving fractional exponents. Simplifying these expressions allows physicists to analyze wave behavior and predict particle interactions more effectively. Similarly, in electrical engineering, analyzing circuit behavior often involves simplifying complex expressions with exponents to determine current and voltage relationships.
