Raise to the power: $\left(-a^3 b^2 c\right)^2$
Answer (1)
Apply the power of a product rule: ( x y ) n = x n y n .
Apply the power of a power rule: ( x m ) n = x mn .
Calculate ( − 1 ) 2 = 1 , ( a 3 ) 2 = a 6 , ( b 2 ) 2 = b 4 , and c 2 .
Multiply the results together to obtain the simplified expression: a 6 b 4 c 2 .
Explanation
Understanding the Problem We are asked to simplify the expression ( − a 3 b 2 c ) 2 . This means we need to raise each factor inside the parentheses to the power of 2. We will use the rules of exponents to achieve this.
Applying the Power of a Product Rule First, we apply the power of a product rule, which states that ( x y ) n = x n y n . In our case, this means: ( − a 3 b 2 c ) 2 = ( − 1 ) 2 ⋅ ( a 3 ) 2 ⋅ ( b 2 ) 2 ⋅ c 2
Applying the Power of a Power Rule Next, we apply the power of a power rule, which states that ( x m ) n = x mn . Applying this rule to the a and b terms, we get:
( a 3 ) 2 = a 3 × 2 = a 6 ( b 2 ) 2 = b 2 × 2 = b 4
Simplifying the Expression Now, we simplify the expression by calculating ( − 1 ) 2 , which is equal to 1. So we have:
( − 1 ) 2 ⋅ ( a 3 ) 2 ⋅ ( b 2 ) 2 ⋅ c 2 = 1 ⋅ a 6 ⋅ b 4 ⋅ c 2 = a 6 b 4 c 2
Final Answer Therefore, the simplified expression is a 6 b 4 c 2 .
Examples
Imagine you are calculating the area of a square where the side length is given by the expression − a 3 b 2 c . Since area cannot be negative, you square the side length to get a positive value. The result, a 6 b 4 c 2 , represents the area of that square. This concept is useful in various fields like physics, where you might deal with squared quantities such as energy or power, ensuring the result is always positive and physically meaningful.
