Simplify this expression.
[tex]12 x^2(3 x^4-2 x)[/tex]
A. [tex]36 x^6-2 x[/tex]
B. [tex]36 x^6-24 x^3[/tex]
C. [tex]36 x^8-2 x[/tex]
D. [tex]36 x^8-24 x^3[/tex]
Answer (2)
Distribute 12 x 2 across the terms in the parentheses: 12 x 2 ( 3 x 4 − 2 x ) = ( 12 x 2 ) ( 3 x 4 ) − ( 12 x 2 ) ( 2 x ) .
Multiply the coefficients and add the exponents for the first term: ( 12 x 2 ) ( 3 x 4 ) = 36 x 6 .
Multiply the coefficients and add the exponents for the second term: ( 12 x 2 ) ( 2 x ) = 24 x 3 .
Combine the terms to get the simplified expression: 36 x 6 − 24 x 3 .
Explanation
Understanding the Problem We are given the expression 12 x 2 ( 3 x 4 − 2 x ) . Our goal is to simplify it by applying the distributive property and combining like terms.
Applying the Distributive Property First, we distribute 12 x 2 to both terms inside the parentheses: 12 x 2 ( 3 x 4 − 2 x ) = ( 12 x 2 ) ( 3 x 4 ) − ( 12 x 2 ) ( 2 x )
Multiplying Coefficients and Adding Exponents Now, we multiply the coefficients and add the exponents of x in each term. For the first term: ( 12 x 2 ) ( 3 x 4 ) = ( 12 × 3 ) ( x 2 × x 4 ) = 36 x 2 + 4 = 36 x 6 For the second term: ( 12 x 2 ) ( 2 x ) = ( 12 × 2 ) ( x 2 × x 1 ) = 24 x 2 + 1 = 24 x 3
Combining Terms Finally, we combine the simplified terms: 36 x 6 − 24 x 3
Final Answer The simplified expression is 36 x 6 − 24 x 3 . Therefore, the correct answer is B.
Examples
Understanding how to simplify expressions like this is useful in many areas, such as physics and engineering, where you might need to manipulate equations to solve for a particular variable. For example, if you're calculating the area of a complex shape or the trajectory of a projectile, you'll often need to simplify algebraic expressions to get to your final answer. Simplifying expressions also helps in computer science when optimizing code, as simpler expressions can lead to faster execution times. Imagine you're designing a game and need to calculate the position of an object; simplifying the equations can make the game run smoother.
The simplified expression is 36 x 6 − 24 x 3 . Therefore, the correct answer is B.
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