Rationalize the Denominator -8/(-6- square root of 7)

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: $-(\frac{8}{-6-\sqrt{7}}\cdot \frac{-6+\sqrt{7}}{-6+\sqrt{7}})$

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 $0.92530998\dots$

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.