Simplify:
[tex]\begin{array}{l}
\frac{x^3}{x^4} \\
\frac{1}{x^{[?]}}
\end{array}[/tex]
Answer (1)
Apply the quotient of powers property: x b x a = x a − b .
Calculate the exponent: 3 − 4 = − 1 .
Rewrite the expression: x − 1 = x 1 1 .
The value of [?] is 1 .
Explanation
Understanding the problem We are given the expression x 4 x 3 which needs to be simplified and expressed in the form x [ ?] 1
Applying the quotient of powers property To simplify the expression x 4 x 3 , we use the quotient of powers property, which states that x b x a = x a − b . In our case, a = 3 and b = 4 .
Calculating the exponent Applying the quotient of powers property, we have: x 4 x 3 = x 3 − 4 = x − 1 The exponent is 3 − 4 = − 1 .
Rewriting the expression Now, we rewrite x − 1 in the form x n 1 . Recall that x − n = x n 1 . Therefore, x − 1 = x 1 1 So, the simplified expression is x 1 1 .
Finding the value of [?] Comparing x 1 1 with x [ ?] 1 we see that the value of [?] is 1. Therefore, the simplified expression is x 1 1 .
Final Answer The simplified expression is x 1 1 , so the value of [?] is 1. 1
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating growth rates or dealing with exponential decay. For example, if a population doubles every year, the population after n years can be expressed as P = P 0 2 n , where P 0 is the initial population. Simplifying such expressions helps in predicting future values and understanding the underlying relationships.
