To determine whether the expression [tex]\(\sqrt{3x}\)[/tex] is equivalent to [tex]\(x \sqrt{3}\)[/tex], let's carefully analyze each mathematical statement.
1. Understanding [tex]\(\sqrt{3x}\)[/tex]:
- The expression [tex]\(\sqrt{3x}\)[/tex] represents the square root of the entire product of [tex]\(3\)[/tex] and [tex]\(x\)[/tex].
- Mathematically, [tex]\(\sqrt{3x}\)[/tex] can be interpreted as [tex]\(\sqrt{3 \cdot x}\)[/tex], which means we first multiply [tex]\(3\)[/tex] by [tex]\(x\)[/tex] and then take the square root of the result.
2. Understanding [tex]\(x\sqrt{3}\)[/tex]:
- The expression [tex]\(x\sqrt{3}\)[/tex] represents the product of [tex]\(x\)[/tex] and the square root of [tex]\(3\)[/tex].
- Here, [tex]\(\sqrt{3}\)[/tex] is computed first and then multiplied by [tex]\(x\)[/tex].
To see why these expressions are not equivalent, consider the following:
- For [tex]\(\sqrt{3x}\)[/tex], the order of operations dictates that multiplication inside the square root occurs first. Thus, it involves different steps relative to the two operations (multiplication then square root).
- For [tex]\(x\sqrt{3}\)[/tex], the square root operation ([tex]\(\sqrt{3}\)[/tex]) is entirely independent of [tex]\(x\)[/tex], and they are multiplied afterward.
To observe this with actual numbers, suppose [tex]\(x = 4\)[/tex]:
- [tex]\(\sqrt{3 \cdot 4} = \sqrt{12}\)[/tex]
- Calculating this: [tex]\(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\)[/tex]
- However, [tex]\(4\sqrt{3}\)[/tex] is simply [tex]\(4\)[/tex] multiplied by [tex]\(\sqrt{3}\)[/tex].
As seen, [tex]\(\sqrt{12}\)[/tex] and [tex]\(4\sqrt{3}\)[/tex] are not the same value, hence [tex]\(\sqrt{3x}\)[/tex] is not equal to [tex]\(x\sqrt{3}\)[/tex].
So, the correct answer is:
B. False
The expressions [tex]\(\sqrt{3 x}\)[/tex] and [tex]\(x \sqrt{3}\)[/tex] are not equivalent.
