$(4 n^3 \cdot n^2)^2$
Answer (1)
First, simplify inside the parenthesis: 4n^3 ^2 = 4n^{3+2} = 4n^5 .
Apply the power of a product rule: ( 4 n 5 ) 2 = 4 2 ( n 5 ) 2 .
Simplify the constants: 4 2 = 16 .
Multiply the exponents: ( n 5 ) 2 = n 52 = n 10 . The final answer is 16 n 10 .
Explanation
Understanding the Problem We are given the expression (4n^3 ^2)^2 . Our goal is to simplify this expression using the rules of exponents.
Simplifying Inside Parentheses First, we simplify the expression inside the parentheses. We have 4n^3 ^2 . When multiplying terms with the same base, we add the exponents. So, n^3 ^2 = n^{3+2} = n^5 . Thus, the expression inside the parentheses becomes 4 n 5 .
Applying the Power of a Product Rule Now, we have ( 4 n 5 ) 2 . To simplify this, we apply the power of a product rule, which states that ( ab ) n = a n b n . In our case, a = 4 , b = n 5 , and n = 2 . So, ( 4 n 5 ) 2 = 4 2 ( n 5 ) 2 .
Simplifying the Expression We know that 4 2 = 16 . Also, when raising a power to a power, we multiply the exponents. So, ( n 5 ) 2 = n 52 = n 10 . Therefore, the expression becomes 16 n 10 .
Final Answer Thus, the simplified expression is 16 n 10 .
Examples
Understanding how to simplify expressions with exponents is crucial in various fields, such as physics and engineering. For instance, when calculating the area of a square with side length 2 x 3 , you would square the side length to get ( 2 x 3 ) 2 = 4 x 6 . This skill is also essential in computer science when analyzing the complexity of algorithms, where exponential notation is frequently used to describe the growth rate of functions.
