Question

To rationalize a denominator of ( (5 - √(7))(9 - √(14)) ), you should multiply the expression by which factor?

a) ( 9 + √(14) )
b) ( 9 + √(7) )
c) ( 9 - √(14) )
d) ( 9 - √(7) )

Answer 1

To rationalize the denominator (5 - √(7))(9 - √(14)), you should multiply by the conjugate of the entire denominator, which is option a) (9 + √(14)).

Step-by-step explanation:

To rationalize a denominator such as (5 - √(7))(9 - √(14)), you need to multiply by a factor that will eliminate the square roots in the denominator. To achieve this, you would typically use the conjugate of the entire denominator. The conjugate of (9 - √(14)) is (9 + √(14)), and multiplying by this will result in a rational denominator. Therefore, the correct factor to multiply with is option a) (9 + √(14)).

Let's demonstrate the process:

1. Multiply the denominator (9 - √(14)) by its conjugate (9 + √(14)).
2. The result will be the difference of squares: 92 - (√(14))2, which simplifies to 81 - 14, or 67.
3. Thus, you have rationalized the denominator and it no longer contains any irrational numbers.