To rationalize a denominator that has more than one term, you typically multiply the fraction by [tex]\(\frac{B}{B}\)[/tex], where [tex]\(B\)[/tex] is actually the conjugate of the denominator, not the numerator.
Let's break down this process through a step-by-step explanation:
1. Identify the denominator: In a fraction where the denominator has more than one term, it's often a binomial expression, such as [tex]\(a + b\sqrt{c}\)[/tex] or [tex]\(a - b\sqrt{c}\)[/tex].
2. Find the conjugate of the denominator: The conjugate of [tex]\(a + b\sqrt{c}\)[/tex] is [tex]\(a - b\sqrt{c}\)[/tex], and the conjugate of [tex]\(a - b\sqrt{c}\)[/tex] is [tex]\(a + b\sqrt{c}\)[/tex].
3. Multiply by the conjugate: To rationalize the denominator, multiply the fraction by 1 in the form of [tex]\(\frac{B}{B}\)[/tex], where [tex]\(B\)[/tex] is the conjugate of the denominator.
For example, if the fraction is [tex]\(\frac{N}{a + b\sqrt{c}}\)[/tex]:
[tex]\[
\frac{N}{a + b\sqrt{c}} \times \frac{a - b\sqrt{c}}{a - b\sqrt{c}}
\][/tex]
4. Simplify by using the difference of squares: When you multiply the denominator [tex]\(a + b\sqrt{c}\)[/tex] by its conjugate [tex]\(a - b\sqrt{c}\)[/tex], you get:
[tex]\[
(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - (b\sqrt{c})^2 = a^2 - b^2c
\][/tex]
This product is a rational number.
5. Final expression: The numerator will also change as it gets multiplied by the conjugate of the denominator, but the crucial part is that the new denominator is rationalized.
Given the above process and clear understanding, the correct answer would be:
B. False
