Rewrite $256^{-\frac{1}{4}}$ with a positive exponent, then simplify.
$256^{-\frac{1}{4}}=\square$
Answer (1)
Rewrite the expression with a positive exponent: 25 6 − 4 1 = 25 6 4 1 1 .
Recognize that 256 = 4 4 , so 25 6 4 1 = ( 4 4 ) 4 1 .
Simplify the exponent: ( 4 4 ) 4 1 = 4 4 × 4 1 = 4 1 = 4 .
Substitute the simplified value: 25 6 4 1 1 = 4 1 . Therefore, the final answer is 4 1 .
Explanation
Understanding the problem We are given the expression 25 6 − 4 1 . Our goal is to rewrite this expression with a positive exponent and then simplify it.
Rewriting with a positive exponent Recall the property of exponents: a − n = a n 1 . Using this property, we can rewrite the given expression as: 25 6 − 4 1 = 25 6 4 1 1 Now we have a positive exponent.
Simplifying the expression Now we need to simplify 25 6 4 1 . This is asking: what number, when raised to the power of 4, equals 256? We can express 256 as a power of 4: 256 = 4 4 . Therefore, 25 6 4 1 = ( 4 4 ) 4 1
Applying exponent rules Using the property of exponents ( a m ) n = a mn , we can simplify the expression further: ( 4 4 ) 4 1 = 4 4 ⋅ 4 1 = 4 1 = 4 So, 25 6 4 1 = 4 .
Final simplification Substitute this value back into the expression we obtained in step 2: 25 6 4 1 1 = 4 1 Therefore, 25 6 − 4 1 = 4 1 .
Examples
Understanding fractional exponents is useful in various fields. For instance, in computer graphics, calculating the n -th root of a number is essential for texture mapping and shading. Also, in finance, compound interest calculations often involve fractional exponents to determine growth rates over fractional periods. This problem demonstrates a fundamental concept applicable in both technical and financial contexts.
