Write the quotient as an exponent: [tex]$y^{10} \div y^7$[/tex]
Answer (1)
Recall the quotient rule of exponents: a m / a n = a m − n .
Apply the quotient rule to the expression: y 10 / y 7 = y 10 − 7 .
Simplify the exponent: 10 − 7 = 3 .
The quotient as an exponent is: y 3 .
Explanation
Understanding the problem We are asked to write the quotient y 10 ÷ y 7 as an exponent. To do this, we need to remember the quotient rule for exponents.
Recalling the quotient rule The quotient rule states that when dividing exponential expressions with the same base, we subtract the exponents. In other words, a m ÷ a n = a m − n , where a is the base and m and n are the exponents.
Applying the rule Applying the quotient rule to the given expression, we have:
y 10 ÷ y 7 = y 10 − 7
Simplifying the exponent Now, we simplify the exponent by subtracting: 10 − 7 = 3 . Therefore, the expression becomes:
y 10 ÷ y 7 = y 3
Final Answer So, the quotient y 10 ÷ y 7 written as an exponent is y 3 .
Examples
Understanding exponential division is crucial in many fields, such as computer science when dealing with data storage or network bandwidth. For example, if a server has a storage capacity represented by 2 20 bytes and each file occupies 2 17 bytes, dividing 2 20 by 2 17 (which simplifies to 2 20 − 17 = 2 3 = 8 ) tells us how many files can be stored on the server. This principle helps in optimizing resource allocation and understanding system limitations.
