e.) $(a+b)^2 \times(a+b)^{-2}$
Answer (1)
1 (if a + b = 0 ).
Explanation
Understanding the Problem We are asked to simplify the expression ( a + b ) 2 " , ( a + b ) − 2 . This involves understanding how exponents work, particularly negative exponents.
Rewriting with Positive Exponents Recall that x − n = x n 1 . Therefore, ( a + b ) − 2 = ( a + b ) 2 1 . We can rewrite the original expression as: ( a + b ) 2 " , ( a + b ) − 2 = ( a + b ) 2 " , ( a + b ) 2 1 .
Multiplying the Terms Now, we can simplify the expression by multiplying: ( a + b ) 2 " , ( a + b ) 2 1 = ( a + b ) 2 ( a + b ) 2 .
Simplifying the Fraction If a + b = 0 , then any non-zero number divided by itself is 1. Therefore, ( a + b ) 2 ( a + b ) 2 = 1 . However, if a + b = 0 , then the expression is undefined because we would be dividing by zero.
Examples
In electrical engineering, when analyzing circuits, you might encounter expressions involving impedance. Simplifying these expressions, which often include terms raised to positive and negative powers, helps engineers determine the overall behavior of the circuit. For example, if ( a + b ) represents a complex impedance, simplifying ( a + b ) 2 × ( a + b ) − 2 helps in understanding the net effect of these impedances in a circuit, ensuring efficient design and troubleshooting.
