e.) $(a+b)^2 \times(a+b)^{-2}$
Answer (1)
Rewrite the expression using the property of exponents: ( a + b ) 2 " , ( a + b ) − 2 = ( a + b ) 2 " , ( a + b ) 2 1 .
Simplify the expression by multiplying the terms: ( a + b ) 2 ( a + b ) 2 .
As long as a + b = 0 , simplify the fraction: ( a + b ) 2 ( a + b ) 2 = 1 .
The simplified expression is 1 .
Explanation
Understanding the Problem We are asked to simplify the expression ( a + b ) 2 " , ( a + b ) − 2 .
Rewriting the Expression To simplify the expression, we can rewrite it using the properties of exponents. Recall that x − n = " , ( 1/ x n ) . Therefore, we can rewrite the expression as: ( a + b ) 2 " , ( a + b ) − 2 = ( a + b ) 2 " , ( a + b ) 2 1
Simplifying the Expression Now, we can simplify the expression by multiplying the terms: ( a + b ) 2 " , ( a + b ) 2 1 = ( a + b ) 2 ( a + b ) 2
Final Simplification As long as a + b = 0 , we can simplify the fraction. Any non-zero number divided by itself is equal to 1. Therefore, ( a + b ) 2 ( a + b ) 2 = 1 If a + b = 0 , then the expression is undefined because we would be dividing by zero.
Examples
Imagine you are simplifying a recipe. If you double the recipe and then halve it, you end up with the original amount. This is similar to the problem where you have an expression raised to a power and then raised to the inverse of that power. Simplifying expressions like this is useful in many areas, such as calculating areas, volumes, and rates of change.
