Simplify: $(-10 a b)^2$
Answer (1)
Apply the power of a product rule: ( − 10 ab ) 2 = ( − 10 ) 2 × a 2 × b 2 .
Calculate ( − 10 ) 2 = 100 .
Substitute the result back into the expression: 100 × a 2 × b 2 .
The simplified expression is 100 a 2 b 2 .
Explanation
Understanding the Problem We are asked to simplify the expression ( − 10 ab ) 2 . This expression involves a product of a constant and two variables, all raised to the power of 2. We will use the power of a product rule to simplify this expression.
Applying the Power of a Product Rule The power of a product rule states that ( x y ) n = x n y n . Applying this rule to our expression, we get: ( − 10 ab ) 2 = ( − 10 ) 2 × a 2 × b 2
Calculating (-10)^2 Now, we need to calculate ( − 10 ) 2 . This means − 10 multiplied by itself: ( − 10 ) 2 = ( − 10 ) × ( − 10 ) = 100
Writing the Simplified Expression Substituting this back into our expression, we have: ( − 10 ab ) 2 = 100 × a 2 × b 2 = 100 a 2 b 2
Examples
Understanding how to simplify expressions with exponents is crucial in various fields, such as physics and engineering. For example, when calculating the area of a square with side length 5 x , you would square the side length to get ( 5 x ) 2 = 25 x 2 . This principle extends to more complex calculations, such as determining the energy stored in a capacitor or the power dissipated in a resistor.
