Simplify the expression: $(2 x)^2 \cdot 3 x^3$
Answer (2)
Apply the power of a product rule: ( 2 x ) 2 = 4 x 2 .
Multiply the result by 3 x 3 : 4 x 2 ⋅ 3 x 3 .
Multiply the coefficients: 4 ⋅ 3 = 12 .
Apply the product of powers rule: x 2 ⋅ x 3 = x 5 . The final simplified expression is 12 x 5 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 2 x ) 2 ⋅ 3 x 3 . This involves using the power of a product rule and the product of powers rule.
Applying the Power of a Product Rule First, we apply the power of a product rule to the term ( 2 x ) 2 . The power of a product rule states that ( ab ) n = a n b n . Therefore, ( 2 x ) 2 = 2 2 ⋅ x 2 = 4 x 2 .
Substituting Back into the Expression Now, we substitute this back into the original expression: 4 x 2 ⋅ 3 x 3 .
Multiplying Coefficients Next, we multiply the coefficients: 4 ⋅ 3 = 12 .
Applying the Product of Powers Rule Then, we apply the product of powers rule to the variable part. The product of powers rule states that x m ⋅ x n = x m + n . Therefore, x 2 ⋅ x 3 = x 2 + 3 = x 5 .
Combining Terms Finally, we combine the coefficient and the variable part to get the simplified expression: 12 x 5 .
Final Answer Therefore, the simplified expression is 12 x 5 .
Examples
Understanding how to simplify expressions like this is useful in many areas of math and science. For example, if you are calculating the area of a rectangle where the sides are expressed in terms of variables, you might need to simplify an expression like this. Suppose the length of a rectangle is ( 2 x ) 2 and the width is 3 x 3 . The area would be ( 2 x ) 2 ⋅ 3 x 3 , which simplifies to 12 x 5 . This skill is also crucial in physics when dealing with polynomial expressions in mechanics or electromagnetism.
The expression ( 2 x ) 2 ⋅ 3 x 3 simplifies to 12 x 5 by applying the power of a product rule and the product of powers rule. First, we calculate ( 2 x ) 2 to get 4 x 2 , then we multiply it by 3 x 3 , yielding 12 x 5 . As a result, the final answer is 12 x 5 .
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