Suppose you have a product like this: (3x)(2.1)(3x)(2.1)
You are asked: Which choice is equivalent to this product when x≠0x \neq 0?
The Correct Answer and Explanation is:
We are given the expression: (3x)(2.1)(3x)(2.1)(3x)(2.1)(3x)(2.1)(3x)(2.1)(3x)(2.1)
We are asked to simplify this product and find which expression is equivalent to it when x≠0x \neq 0x=0.
Step 1: Group like terms
The expression contains two instances of 3x3x3x and two instances of 2.12.12.1, so we can group them together: (3x)(3x)(2.1)(2.1)(3x)(3x)(2.1)(2.1)(3x)(3x)(2.1)(2.1)
Now, use the associative property of multiplication to rearrange: (3x⋅3x)⋅(2.1⋅2.1)(3x \cdot 3x) \cdot (2.1 \cdot 2.1)(3x⋅3x)⋅(2.1⋅2.1)
Step 2: Multiply constants and variables
Start with 3x⋅3x3x \cdot 3x3x⋅3x: 3x⋅3x=(3⋅3)(x⋅x)=9x23x \cdot 3x = (3 \cdot 3)(x \cdot x) = 9x^23x⋅3x=(3⋅3)(x⋅x)=9×2
Then multiply 2.1⋅2.12.1 \cdot 2.12.1⋅2.1: 2.1⋅2.1=4.412.1 \cdot 2.1 = 4.412.1⋅2.1=4.41
Step 3: Final multiplication
Now multiply both parts together: 9×2⋅4.41=(9⋅4.41)x29x^2 \cdot 4.41 = (9 \cdot 4.41)x^29×2⋅4.41=(9⋅4.41)x2
Calculate 9⋅4.419 \cdot 4.419⋅4.41: 9⋅4.41=39.699 \cdot 4.41 = 39.699⋅4.41=39.69
So the fully simplified expression is: 39.69×2\boxed{39.69x^2}39.69×2
Why this is equivalent when x≠0x \neq 0x=0
The expression is valid for all real values of xxx except where the expression might be undefined, such as division by zero. However, this is a multiplication problem and does not involve any denominators or division. So technically, it’s defined for all real xxx, including zero.
But the question says “when x≠0x \neq 0x=0,” probably to emphasize we’re not dealing with an identity that relies on division. Since the expression simplifies without any restrictions on xxx, and we’re not dividing by xxx, this condition does not affect the simplification.
So, the final answer is: 39.69×2\boxed{39.69x^2}39.69×2