Which expression is equivalent to $\left(x^{\frac{1}{4}} y^{16}\right)^{\frac{1}{2}}$ ?
A. $x^{\frac{1}{2}} y^4$
B. $x^{\frac{1}{8}} y^8$
C. $x^{\frac{1}{4}} y^8$
D. $x^{\frac{1}{4}} y^4$
Answer (2)
Apply the power of a product rule: ( ab ) n = a n b n .
Apply the power of a power rule: ( a m ) n = a mn .
Simplify the exponents: 4 1 ⋅ 2 1 = 8 1 and 16 ⋅ 2 1 = 8 .
The equivalent expression is x 8 1 y 8 .
Explanation
Understanding the problem We are given the expression ( x 4 1 y 16 ) 2 1 and asked to find an equivalent expression. We will use the properties of exponents to simplify the given expression.
Applying the power of a product rule We will use the power of a product rule, which states that ( ab ) n = a n b n . Applying this rule, we get ( x 4 1 y 16 ) 2 1 = ( x 4 1 ) 2 1 ( y 16 ) 2 1
Applying the power of a power rule Next, we use the power of a power rule, which states that ( a m ) n = a mn . Applying this rule to each term, we have ( x 4 1 ) 2 1 = x 4 1 ⋅ 2 1 ( y 16 ) 2 1 = y 16 ⋅ 2 1
Simplifying the exponents Now, we simplify the exponents: 4 1 ⋅ 2 1 = 8 1 16 ⋅ 2 1 = 8 So we have x 8 1 y 8
Final Answer Therefore, the expression equivalent to ( x 4 1 y 16 ) 2 1 is x 8 1 y 8 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating growth rates, dealing with scientific notation in physics, or even in computer science when analyzing algorithms. For example, if you are analyzing the time complexity of an algorithm and find it to be O (( n 2 1 ) 2 ) , simplifying this expression to O ( n ) helps you understand that the algorithm's runtime grows linearly with the input size n .
The expression ( x 4 1 y 16 ) 2 1 simplifies to x 8 1 y 8 . The correct option is B.
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