Simplify the expression $\left(\frac{x^3}{x^4}\right)^2$.
A) $\frac{1}{x^2}$
B) $\frac{1}{2}$
C) $\frac{1}{x}$
D) $\frac{x}{x^2}$
Answer (1)
Simplify the fraction inside the parentheses using the rule x b x a = x a − b , which gives x 4 x 3 = x − 1 = x 1 .
Raise the simplified fraction to the power of 2: ( x 1 ) 2 = x 2 1 2 = x 2 1 .
The simplified expression is x 2 1 .
The final answer is x 2 1 .
Explanation
Understanding the Problem We are given the expression ( x 4 x 3 ) 2 . Our goal is to simplify this expression using the properties of exponents.
Simplifying Inside Parentheses First, we simplify the fraction inside the parentheses. Recall the rule for dividing exponential terms with the same base: x b x a = x a − b . Applying this rule, we have x 4 x 3 = x 3 − 4 = x − 1 Since x − 1 = x 1 , we can rewrite the expression as ( x 4 x 3 ) 2 = ( x 1 ) 2
Applying the Exponent Next, we raise the simplified fraction to the power of 2. Recall the rule ( a / b ) n = a n / b n . Applying this rule, we get ( x 1 ) 2 = x 2 1 2 = x 2 1
Final Answer Therefore, the simplified expression is x 2 1 . Comparing this with the given options, we see that it matches option A.
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating areas and volumes. For example, if you have a square with side length x 3 and you want to find the area of a similar square with sides scaled down by a factor of x , you would need to simplify ( x x 3 ) 2 = ( x 2 ) 2 = x 4 to find the area of the smaller square.
