Simplify $(3 x)^{-1}$.
A) $4 x^6$
B) $\frac{1}{x^6}$
C) $81 x^4$
D) $\frac{1}{3 x}$
Answer (1)
Apply the negative exponent rule: ( 3 x ) − 1 = ( 3 x ) 1 1 .
Simplify the denominator: ( 3 x ) 1 = 3 x .
The simplified expression is 3 x 1 .
The correct option is 3 x 1 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 3 x ) − 1 . This involves understanding how negative exponents work.
Applying the Negative Exponent Rule Recall that a negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words, a − n = a n 1 . Applying this to our expression, we have ( 3 x ) − 1 = ( 3 x ) 1 1 .
Simplifying the Expression Now, we simplify the denominator. Since ( 3 x ) 1 is just 3 x , our expression becomes 3 x 1 .
Choosing the Correct Option Comparing our simplified expression 3 x 1 with the given options, we see that it matches option D. Therefore, the correct answer is D.
Examples
Understanding negative exponents is crucial in various fields like physics and engineering. For instance, in physics, the gravitational force between two objects is inversely proportional to the square of the distance between them, represented as F = G r 2 m 1 m 2 = G m 1 m 2 r − 2 . Simplifying expressions with negative exponents helps in calculating and understanding such relationships effectively. Similarly, in finance, understanding exponential decay, which involves negative exponents, is essential for calculating depreciation or the present value of future cash flows.
