$(4 n^3 \cdot n^2)^2$
Answer (1)
First, simplify the expression inside the parentheses by adding the exponents: 4 n^3 ^2 = 4n^5 .
Then, apply the outer exponent to both the constant and the variable: ( 4 n 5 ) 2 = 4 2 ( n 5 ) 2 .
Simplify the constant and the variable: 4 2 = 16 and ( n 5 ) 2 = n 10 .
The final simplified expression is 16 n 10 .
Explanation
Understanding the problem We are asked to simplify the expression \left(4 n^3 ^2\right)^2 . To do this, we will first simplify the expression inside the parentheses and then apply the outer exponent.
Simplifying inside the parentheses First, we simplify the expression inside the parentheses. We have 4 n^3 ^2 . Since we are multiplying terms with the same base ( n ), we add the exponents: n^3 ^2 = n^{3+2} = n^5 . So, the expression inside the parentheses becomes 4 n 5 .
Applying the outer exponent Now, we have to square the expression 4 n 5 . This means we raise both the constant and the variable term to the power of 2: (4n^5)^2 = 4^2 ^{5 2} .
Final simplification We know that 4 2 = 44 = 16 and n 52 = n 10 . Therefore, the simplified expression is 16 n 10 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating the area or volume of geometric shapes. For example, if you have a square with side length 2 x 3 , its area would be ( 2 x 3 ) 2 = 4 x 6 . This type of simplification is also crucial in physics when dealing with quantities that scale with powers, such as gravitational force or electric fields.
