Question

√ x 2 = x 2 4 = x 1 2 = √ x Based on this, the student claims that 4 √ x 2 = √ x for all values of x.

Answer 1

The claim that 4√x^2 = √x for all values of x is incorrect.

Step-by-step explanation:

It is not true that √x^2 = x^(2/4) = x^(1/2) = √x for all values of x. This claim is incorrect.

Let's analyze each step:

1. √x^2 = x^(2/4)
2. x^(2/4) = x^(1/2)
3. x^(1/2) = √x

Even though these steps might seem valid, it's important to consider the properties of exponents. In the second step, x^(2/4) does not equal x^(1/2). They are different expressions. The original equation, therefore, cannot be simplified to 4√x^2 = √x for all values of x.

Answer 2

The student's claim is incorrect. While it is true that √x^2 = x for all non-negative values of x, this property does not apply to expressions with multiple terms, such as 4√x^2.

Step-by-step explanation:

consider the case where x = -1. Then, √x^2 = √(-1)^2 = 1, but 4√x^2 = 4√1 = 4. On the other hand, √x = √(-1) is undefined, so the equation 4√x^2 = √x is not true for this value of x.

Therefore, the student's claim is not true for all values of x, and it is important to be careful when applying algebraic properties to expressions with multiple terms.