$\left(6 b^2\right)^2$
Answer (1)
Apply the power of a product rule: ( 6 b 2 ) 2 = 6 2 × ( b 2 ) 2 .
Simplify 6 2 = 36 .
Apply the power rule: ( b 2 ) 2 = b 2 × 2 = b 4 .
The simplified expression is 36 b 4 .
Explanation
Understanding the Problem We are given the expression ( 6 b 2 ) 2 and we need to simplify it.
Applying the Power of a Product Rule To simplify the expression, we will use the power of a product rule, which states that ( ab ) n = a n b n . In our case, a = 6 , b = b 2 , and n = 2 . So, we have ( 6 b 2 ) 2 = 6 2 × ( b 2 ) 2
Simplifying the Terms Next, we need to simplify 6 2 and ( b 2 ) 2 . We know that 6 2 = 6 × 6 = 36 . For ( b 2 ) 2 , we use the power rule, which states that ( a m ) n = a mn . In our case, a = b , m = 2 , and n = 2 . So, we have ( b 2 ) 2 = b 2 × 2 = b 4
Final Simplification Now, we substitute these simplified terms back into our expression: ( 6 b 2 ) 2 = 6 2 × ( b 2 ) 2 = 36 × b 4 = 36 b 4
Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as calculating areas and volumes. For example, if you have a square with side length 6 b 2 , its area would be ( 6 b 2 ) 2 = 36 b 4 . This skill is also fundamental in physics, where you might need to calculate the kinetic energy of an object, which involves squaring its velocity.
