Which expression is equivalent to $\left(64 y^{100}\right)^{\frac{1}{2}}$?
A. $8 y^{10}$
B. $8 y^{50}$
C. $32 y^{10}$
D. $32 y^{50}$
Answer (2)
Apply the power of a product rule: ( 64 y 100 ) 2 1 = 6 4 2 1 × ( y 100 ) 2 1 .
Calculate the square root of 64: 6 4 2 1 = 8 .
Apply the power of a power rule: ( y 100 ) 2 1 = y 100 × 2 1 = y 50 .
Combine the results: The equivalent expression is 8 y 50 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to ( 64 y 100 ) 2 1 . This involves applying the exponent 2 1 to both the constant 64 and the variable y 100 .
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 . In our case, this means ( 64 y 100 ) 2 1 = 6 4 2 1 ⋅ ( y 100 ) 2 1 .
Calculating the Square Root of 64 First, let's calculate 6 4 2 1 . This is the same as finding the square root of 64, which is 8 since 8 × 8 = 64 .
Applying the Power of a Power Rule Next, we need to simplify ( y 100 ) 2 1 . We use the power of a power rule, which states that ( a m ) n = a m ⋅ n . Therefore, ( y 100 ) 2 1 = y 100 ⋅ 2 1 = y 50 .
Combining the Results Now, we combine the results: 6 4 2 1 ⋅ ( y 100 ) 2 1 = 8 ⋅ y 50 = 8 y 50 .
Final Answer Therefore, the expression equivalent to ( 64 y 100 ) 2 1 is 8 y 50 .
Examples
Imagine you are designing a square garden and want to know the length of each side if you know the area. If the area of the garden is represented by the expression 64 y 100 , then the side length would be the square root of the area, which simplifies to 8 y 50 . This type of simplification is useful in various fields, including engineering, physics, and computer science, where complex expressions need to be simplified for easier calculations and understanding.
The equivalent expression to ( 64 y 100 ) 2 1 is 8 y 50 , achieved by applying the power of a product rule and simplifying the terms. Thus, the correct answer is option B. 8 y 50 .
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