Simplify the expression. [tex]$\frac{x^{-\frac{1}{2}}}{x^{-\frac{1}{4}}}$[/tex]
Write your answer using only positive exponents. Assume that all variables are positive real numbers.
Answer (2)
Apply the quotient of powers property: x − 4 1 x − 2 1 = x − 2 1 − ( − 4 1 ) .
Simplify the exponent: − 2 1 + 4 1 = − 4 1 .
Rewrite the expression with a positive exponent: x − 4 1 = x 4 1 1 .
The simplified expression is x 4 1 1 .
Explanation
Understanding the Problem We are given the expression x − 4 1 x − 2 1 and asked to simplify it, writing the answer using only positive exponents. We assume that x is a positive real number.
Applying the Quotient of Powers Property To simplify the expression, we use the quotient of powers property, which states that a n a m = a m − n . In our case, a = x , m = − 2 1 , and n = − 4 1 . Applying this property, we have x − 4 1 x − 2 1 = x − 2 1 − ( − 4 1 ) = x − 2 1 + 4 1
Simplifying the Exponent Now, we simplify the exponent: − 2 1 + 4 1 = − 4 2 + 4 1 = − 4 1 So the expression becomes x − 4 1 .
Using Positive Exponents Since we need to write the answer using only positive exponents, we rewrite x − 4 1 as x 4 1 1 .
Final Answer Therefore, the simplified expression with positive exponents is x 4 1 1 .
Examples
Understanding exponents and their properties is crucial in various fields, such as physics and engineering. For example, when dealing with wave propagation, the intensity of a wave decreases with the square of the distance from the source. This relationship can be expressed using exponents, and simplifying such expressions helps in calculating the intensity at different distances. Similarly, in finance, compound interest calculations involve exponents, and simplifying these expressions can help in determining the future value of an investment.
To simplify the expression x − 4 1 x − 2 1 , we apply the quotient of powers property, subtract the exponents, and simplify to get x − 4 1 = x 4 1 1 . The final answer is x 4 1 1 .
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