Simplify the expression below.
$(2 x)^4$
A. $2 x^4$
B. $6 x^4$
C. $8 x^4$
D. $16 x^4$
Answer (1)
Apply the power of a product rule: $(2x)^4 = 2^4
Calculate 2 4 = 16 .
Simplify the expression: ( 2 x ) 4 = 16 x 4 .
The simplified expression is 16 x 4 .
Explanation
Understanding the Expression We are asked to simplify the expression ( 2 x ) 4 . This involves applying the power of a product rule, which states that ( ab ) n = a n b n . In our case, a = 2 , b = x , and n = 4 .
Applying the Power of a Product Rule Applying the power of a product rule, we get ( 2 x ) 4 = 2 4 × x 4 . Now we need to calculate 2 4 .
Calculating the Power We calculate 2 4 as 2 × 2 × 2 × 2 = 16 . Therefore, ( 2 x ) 4 = 16 x 4 .
Identifying the Correct Option Comparing our simplified expression 16 x 4 with the given options, we see that it matches option D.
Final Answer Therefore, the simplified expression is 16 x 4 . The correct answer is D.
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 , the area is ( 2 x ) 2 = 4 x 2 . Similarly, the volume of a cube with side length 2 x is ( 2 x ) 3 = 8 x 3 . These calculations are fundamental in determining physical quantities and designing structures.
