Rationalize the Denominator -8/(-6- square root of 7)
The problem given is from the field of algebra and involves the process known as rationalizing the denominator. This process is typically applied to a mathematical expression or fraction where the denominator contains a square root or another irrational number. The goal of rationalizing the denominator is to eliminate any radicals or irrational numbers from the bottom of the fraction, leaving it in a form considered more "rational" or simpler to work with.
Specifically, the question provides a fraction, -8/(-6 - √7), which has a denominator that includes a subtraction operation between a negative integer and the square root of a positive integer (7). The task is to manipulate this expression so that the resulting denominator is a rational number, free of any square roots. This often involves multiplying the numerator and the denominator by a conjugate or some form of the denominator's terms that will help eliminate the square root when simplified.
$\frac{- 8}{- 6 - \sqrt{7}}$
Answer
Solution:
Step 1:
Extract the negative sign from the fraction to simplify the expression: $- \frac{8}{-6 - \sqrt{7}}$
Step 2:
Multiply the fraction by the conjugate of the denominator to eliminate the square root: $- \left( \frac{8}{-6 - \sqrt{7}} \cdot \frac{-6 + \sqrt{7}}{-6 + \sqrt{7}} \right)$
Step 3:
Apply the multiplication to both the numerator and the denominator: $- \frac{8(-6 + \sqrt{7})}{(-6 - \sqrt{7})(-6 + \sqrt{7})}$
Step 4:
Use the difference of squares formula to expand the denominator: $- \frac{8(-6 + \sqrt{7})}{36 - (\sqrt{7})^2}$
Step 5:
Simplify the denominator by performing the subtraction: $- \frac{8(-6 + \sqrt{7})}{29}$
Step 6:
Distribute the negative sign within the numerator: $- \frac{8(-1(6) + \sqrt{7})}{29}$
Step 7:
Factor out the negative sign from the terms in the numerator: $- \frac{8(-1(6) -1(- \sqrt{7}))}{29}$
Step 8:
Simplify the expression by combining the negative signs: $- \frac{8(-1(6 - \sqrt{7}))}{29}$
Step 9:
Final simplification steps:
Step 9.1:
Remove the double negative in the fraction: $-(- \frac{8(6 - \sqrt{7})}{29})$
Step 9.2:
Multiply the two negative signs to get a positive result: $1 \cdot \frac{8(6 - \sqrt{7})}{29}$
Step 9.3:
Multiply the fraction by 1 to maintain its value: $\frac{8(6 - \sqrt{7})}{29}$
Step 10:
Present the result in different forms:
- Exact Form: $\frac{8(6 - \sqrt{7})}{29}$
- Decimal Form: Approximately $0.92530998 \ldots$
Knowledge Notes:
Rationalizing the Denominator: This process involves eliminating the square root or irrational number from the denominator of a fraction. This is done by multiplying the fraction by a form of 1 that will clear the radical without changing the value of the expression.
Conjugate: 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 a rational number.
Difference of Squares: This is a pattern used in algebra where $(a + b)(a - b) = a^2 - b^2$. It is useful for simplifying expressions where a binomial is multiplied by its conjugate.
Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses. It is expressed as $a(b + c) = ab + ac$.
Simplifying Expressions: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.
