g) $y^7-\frac{1}{y^5}$
Answer (1)
y 5 y 12 − 1
Explanation
Understanding the Problem We are asked to simplify the expression y 7 − y 5 1 . This involves combining the terms into a single fraction.
Finding a Common Denominator To combine the terms, we need a common denominator. The common denominator for y 7 and y 5 1 is y 5 . We rewrite y 7 as y 5 y 7 ⋅ y 5 = y 5 y 7 + 5 = y 5 y 12 .
Rewriting the Expression Now we can rewrite the original expression as: y 7 − y 5 1 = y 5 y 12 − y 5 1
Combining the Fractions Combining the fractions, we get: y 5 y 12 − 1
Final Simplified Expression The expression is now simplified to a single fraction. The numerator y 12 − 1 can be factored further as a difference of squares, but it's not necessary for the simplification. Thus, the simplified expression is y 5 y 12 − 1 .
Examples
In algebra, simplifying expressions like this is crucial for solving equations and understanding functions. For example, if you're analyzing the behavior of a function involving terms like y 7 and y 5 1 , simplifying it to y 5 y 12 − 1 can make it easier to find roots, determine asymptotes, or analyze its derivative. This skill is also useful in physics, where you might encounter similar expressions when dealing with forces or potentials that depend on distance.
