Question

A student incorrectly claims that 5 + √2 = 5 + √5. How could you convince the student that a mistake has been made? How would you explain the correct way of rationalizing the denominator?

A) Explain that 5 + √2 is not equal to 5 + √5, and rationalizing the denominator involves multiplying by the conjugate.
B) Agree with the student's claim and tell them that 5 + √2 and 5 + √5 are equivalent.
C) Provide examples and visual aids to demonstrate the difference between the two expressions.
D) Suggest seeking help from a math teacher to clarify the concept of rationalizing the denominator.

Answer 1

5 + √2 is unequal to 5 + √5 since square roots represent unique numbers based on the value they produce when multiplied by themselves. Rationalizing the denominator requires multiplying both numerator and denominator by the conjugate of the denominator to eliminate the square root.

Step-by-step explanation:

To demonstrate that 5 + √2 is not equal to 5 + √5, we can use the fact that square roots represent the value that, when multiplied by itself, equals the number under the root sign. For instance, √2 multiplied by itself equals 2, and √5 multiplied by itself equals 5. These two values are clearly not the same, thus the initial expressions are different. Furthermore, the process of rationalizing the denominator involves eliminating the square root in the denominator. If we have a fraction with a square root in the denominator, like 1/√2, to rationalize it we would multiply both numerator and denominator by the conjugate, which for a single term like √2 would just be √2. So, we would get (1√2)/(√2√2) which simplifies to √2/2.