Question

To rationalize the denominator of 5− √7/9-√14 , you should multiply the expression by which fraction?

A. 9-14/9+√14
B. 9+ √14/9+ √14
C. Both a and b
D. None of the above

Answer 1

To rationalize the denominator of
`\((5 - √(7))/(9 - √(14))\),` you should multiply the expression by the conjugate of the denominator, which is
`\(9 + √(14)/9 + √(14)\)` (Option B).

Step-by-step explanation:

To rationalize the denominator, we multiply the expression by the conjugate of the denominator. The conjugate of
`\(9 - √(14)\) is \(9 + √(14)\).` Multiplying the expression by this conjugate eliminates the square root in the denominator, resulting in a rationalized expression.

The multiplication is done as follows:

`\[ (5 - √(7))/(9 - √(14)) * (9 + √(14))/(9 + √(14)) \]`

This results in:

`\[ ((5 - √(7))(9 + √(14)))/((9 - √(14))(9 + √(14))) \]`

Expanding the numerator and denominator further:

`\[ (45 + 5√(14) - 9√(7) - √(98))/(81 - 14) \]`

Simplifying the expression gives the final result:

`\[ (45 - 9√(7) + 5√(14) - √(98))/(67) \]`

Therefore, multiplying by the conjugate effectively rationalizes the denominator.

The provided options (A, B, C, D) suggest different choices for the conjugate, and the correct one is
`\(9 + √(14)/9 + √(14)\)` (Option B), which is the conjugate of the original denominator.