Simplify the expression: $y^n \div y^4$
Answer (2)
Apply the quotient of powers property: a n a m = a m − n .
Apply the property to the expression: y 4 y n = y n − 4 .
The simplified expression is: y n − 4 .
Explanation
Understanding the Problem We are given the expression y n ÷ y 4 . Our goal is to simplify it using the properties of exponents.
Applying the Quotient of Powers Property To simplify the expression, we will use the quotient of powers property, which states that when dividing exponential expressions with the same base, we subtract the exponents: a n a m = a m − n In our case, the base is y , and the exponents are n and 4 .
Simplifying the Expression Applying the quotient of powers property to the given expression, we have: y 4 y n = y n − 4 So, the simplified expression is y n − 4 .
Examples
Understanding exponential expressions and their simplification is crucial in various fields, such as calculating compound interest, modeling population growth, and analyzing radioactive decay. For example, if a population of bacteria doubles every hour, the population at time t can be modeled as P ( t ) = P 0 × 2 t , where P 0 is the initial population. Simplifying such expressions helps in predicting future population sizes and understanding the dynamics of growth.
To simplify the expression y n ÷ y 4 , we use the quotient of powers property, which results in y n − 4 . Therefore, the simplified expression is y n − 4 . This method of simplification is an essential skill in algebra involving exponents.
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