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)
The equivalent expression to ( x 4 1 y 16 ) 2 1 is x 8 1 y 8 , which corresponds to option B. By applying the rules of exponents, we simplified the original expression step by step. Therefore, the answer is option B: x 8 1 y 8 .
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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 expression: ( x 4 1 y 16 ) 2 1 = x 8 1 y 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 . Our goal is to simplify this expression using the properties of exponents.
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 will use the power of a power rule, which states that ( a m ) n = a mn . Applying this rule to the first term, we have ( x 4 1 ) 2 1 = x 4 1 ⋅ 2 1 = x 8 1 Applying the power of a power rule to the second term, we have ( y 16 ) 2 1 = y 16 ⋅ 2 1 = y 8
Combining the Results Now, we combine the simplified terms: x 8 1 y 8 Thus, the equivalent expression is x 8 1 y 8 .
Final Answer The expression equivalent to ( x 4 1 y 16 ) 2 1 is x 8 1 y 8 .
Examples
Understanding exponents is crucial in many fields, such as computer science and physics. For example, in computer graphics, transformations like scaling and rotations often involve matrix operations with exponents. In physics, exponential functions are used to model radioactive decay or population growth. Simplifying expressions with exponents helps in making these models easier to understand and manipulate.
