$(2 p^3 \cdot 2 p)^2$
Answer (1)
First, multiply the terms inside the parenthesis: 2 p 3 × 2 p = 4 p 4 .
Then, square the resulting expression: ( 4 p 4 ) 2 = 16 p 8 .
The simplified expression is 16 p 8 .
Explanation
Simplifying Inside Parentheses Let's simplify the expression step-by-step. First, we need to simplify the expression inside the parentheses. We have 2 p 3 × 2 p . To simplify this, we multiply the coefficients and add the exponents of p .
Multiplying Coefficients and Adding Exponents Multiplying the coefficients, we have 2 × 2 = 4 . Adding the exponents of p , we have p 3 × p = p 3 + 1 = p 4 . So, the expression inside the parentheses simplifies to 4 p 4 .
Squaring the Expression Now we need to square the simplified expression, which is ( 4 p 4 ) 2 . To do this, we square the coefficient and multiply the exponent of p by 2.
Final Simplification Squaring the coefficient, we have 4 2 = 16 . Multiplying the exponent of p by 2, we have ( p 4 ) 2 = p 4 × 2 = p 8 . Therefore, the final simplified expression is 16 p 8 .
Final Answer So, ( 2 p 3 × 2 p ) 2 = 16 p 8 .
Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as calculating the area or volume of geometric shapes. For instance, if you have a square with side length 2 p 3 × 2 p , the area would be ( 2 p 3 × 2 p ) 2 , which simplifies to 16 p 8 . This skill is also fundamental in physics, where you might need to calculate the kinetic energy of an object, which involves squaring velocity terms.
