Which is equivalent to $64^{\frac{1}{4}}$ ?
$2 \sqrt[4]{4}$
4
16
$16 \sqrt[4]{4}$
Answer (1)
Simplify the expression: 6 4 4 1 = ( 2 6 ) 4 1 = 2 2 3 .
Rewrite the simplified expression: 2 2 3 = 2 1 + 2 1 = 2 2 .
Compare the simplified expression with the choices: 2 4 4 = 2 ( 4 4 1 ) = 2 ( 2 2 1 ) = 2 2 .
The equivalent expression is: 2 4 4
Explanation
Understanding the Problem We are given the expression 6 4 4 1 and asked to find an equivalent expression from the given choices.
Listing the Choices The choices are: 2 4 4 4
16 16 4 4
Simplifying the Expression We want to simplify the expression 6 4 4 1 . We can rewrite 64 as 2 6 , so the expression becomes ( 2 6 ) 4 1 = 2 4 6 = 2 2 3 .
Further Simplification We can rewrite 2 2 3 as 2 1 + 2 1 = 2 1 ⋅ 2 2 1 = 2 2 .
Comparing with Choices Now, let's rewrite the given choices to compare with 2 2 .
Choice 1: 2 4 4 = 2 ( 4 4 1 ) = 2 ( 2 2 ) 4 1 = 2 ( 2 4 2 ) = 2 ( 2 2 1 ) = 2 2 .
Choice 2: 4 Choice 3: 16 Choice 4: 16 4 4 = 16 ( 4 4 1 ) = 16 ( 2 2 ) 4 1 = 16 ( 2 2 1 ) = 16 2
Finding the Equivalent Expression Comparing the simplified expression 2 2 with the choices, we see that 2 4 4 is equivalent to 2 2 .
Final Answer Therefore, the equivalent expression to 6 4 4 1 is 2 4 4 .
Examples
Imagine you are calculating the side length of a square garden with an area of 64 square feet, and you need to divide the garden into four equal smaller squares. Finding 6 4 4 1 helps determine the side length of each of these smaller squares. This concept is useful in various scenarios, such as scaling designs, calculating dimensions in architecture, or determining growth rates in biological studies. Understanding fractional exponents allows for precise calculations in fields requiring proportional scaling and dimensional analysis.
