Rationalize the Denominator -6/(-7- square root of 13)
The problem is asking for the process of rationalizing the denominator in the given fractional expression. In this case, the denominator is a binomial containing an irrational number (the square root of 13) and an integer (-7). Rationalizing the denominator means to eliminate any square roots or other radicals from the denominator so that it becomes a rational number. This often involves multiplying the numerator and denominator by a conjugate of the denominator. The conjugate of a binomial a + b is a - b; similarly, the conjugate of a - b is a + b. In this problem, to rationalize the denominator of -6/(-7 - √13), one would multiply both the numerator and the denominator by the conjugate of the denominator, which is (-7 + √13). The goal is to obtain a denominator that is a rational number without changing the value of the original expression.
$\frac{- 6}{- 7 - \sqrt{13}}$
Answer
Solution:
Step 1:
Extract the negative sign from the fraction to obtain $- \frac{6}{-7 - \sqrt{13}}$.
Step 2:
Multiply the fraction by $\frac{-7 + \sqrt{13}}{-7 + \sqrt{13}}$ to rationalize the denominator: $- \left( \frac{6}{-7 - \sqrt{13}} \cdot \frac{-7 + \sqrt{13}}{-7 + \sqrt{13}} \right)$.
Step 3:
Perform the multiplication: $- \frac{6(-7 + \sqrt{13})}{(-7 - \sqrt{13})(-7 + \sqrt{13})}$.
Step 4:
Use the difference of squares to expand the denominator: $- \frac{6(-7 + \sqrt{13})}{49 - (\sqrt{13})^2}$.
Step 5:
Simplify the denominator: $- \frac{6(-7 + \sqrt{13})}{36}$.
Step 6:
Reduce the fraction by canceling out common factors.
Step 6.1:
Factor out a 6 from the denominator: $- \frac{6(-7 + \sqrt{13})}{6 \cdot 6}$.
Step 6.2:
Cancel the 6 in the numerator and denominator: $- \frac{\cancel{6}(-7 + \sqrt{13})}{\cancel{6} \cdot 6}$.
Step 6.3:
Rewrite the simplified expression: $- \frac{-7 + \sqrt{13}}{6}$.
Step 7:
Rewrite $-7$ as $-1(7)$: $- \frac{-1(7) + \sqrt{13}}{6}$.
Step 8:
Factor out a $-1$ from the numerator: $- \frac{-1(7) - 1(-\sqrt{13})}{6}$.
Step 9:
Factor $-1$ out of the entire numerator: $- \frac{-1(7 - \sqrt{13})}{6}$.
Step 10:
Simplify the expression.
Step 10.1:
Remove the double negative in front of the fraction: $-- \frac{7 - \sqrt{13}}{6}$.
Step 10.2:
Multiply $-1$ by $-1$ to get $1$: $1 \cdot \frac{7 - \sqrt{13}}{6}$.
Step 10.3:
Multiply the fraction by $1$: $\frac{7 - \sqrt{13}}{6}$.
Step 11:
Present the result in various forms.
Exact Form: $\frac{7 - \sqrt{13}}{6}$
Decimal Form: Approximately $0.56574145 \ldots$
Knowledge Notes:
To rationalize the denominator means to eliminate any irrational numbers (like square roots) from the denominator of a fraction. This is often done by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial $a + b$ is $a - b$. When a binomial is multiplied by its conjugate, the result is a difference of squares, which is always a rational number.
In this problem, the conjugate of $-7 - \sqrt{13}$ is $-7 + \sqrt{13}$. Multiplying the denominator by its conjugate results in the difference of squares, which simplifies to a rational number. The numerator is also affected by this multiplication, and it must be simplified as well.
The FOIL method stands for First, Outer, Inner, Last, and it is a technique used to expand the product of two binomials. In the context of rationalizing the denominator, it is used to expand the product of the denominator and its conjugate.
Finally, it is important to simplify the resulting expression by canceling out any common factors in the numerator and denominator to arrive at the simplest form of the fraction.
