Simplify: $\frac{-15\left(y^2\right)^5}{-5 y^2 y^3}$
Answer (1)
Simplify the numerator using the power of a power rule: ( y 2 ) 5 = y 10 .
Simplify the denominator using the product of powers rule: y 2 y 3 = y 5 .
Divide the coefficients: − 5 − 15 = 3 .
Divide the variables using the quotient of powers rule: y 5 y 10 = y 5 . The simplified expression is 3 y 5 .
Explanation
Understanding the Problem We are given the expression − 5 y 2 y 3 − 15 ( y 2 ) 5 and asked to simplify it. Let's break it down step by step!
Simplifying the Numerator First, we simplify the numerator. We have ( y 2 ) 5 . Using the power of a power rule, which states that ( a m ) n = a m × n , we get ( y 2 ) 5 = y 2 × 5 = y 10 . So the numerator becomes − 15 y 10 .
Simplifying the Denominator Next, we simplify the denominator. We have − 5 y 2 y 3 . Using the product of powers rule, which states that a m × a n = a m + n , we get y 2 y 3 = y 2 + 3 = y 5 . So the denominator becomes − 5 y 5 .
Rewriting the Expression Now, we rewrite the expression as − 5 y 5 − 15 y 10 . To simplify this, we divide the coefficients and the variables separately.
Dividing the Coefficients Dividing the coefficients, we have − 5 − 15 = 3 .
Dividing the Variables Dividing the variables, we have y 5 y 10 . Using the quotient of powers rule, which states that a n a m = a m − n , we get y 5 y 10 = y 10 − 5 = y 5 .
Combining the Results Finally, we combine the results to get the simplified expression: 3 y 5 .
Examples
Imagine you're organizing files on your computer. Simplifying expressions like this is similar to cleaning up a folder with many subfolders and files. By applying exponent rules, you can efficiently consolidate and manage your files, making them easier to locate and use. This is useful in computer science, where simplifying expressions can optimize code and improve performance.
