Answered

Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Rationalize the denominator of [tex]\frac{26}{4+\sqrt{3}}[/tex].

Give your answer in its simplest form, with any brackets expanded.

Sagot :

GinnyAnswer
To rationalize the denominator of [tex]\(\frac{26}{4 + \sqrt{3}}\)[/tex], we need to eliminate the square root from the denominator. To achieve this, we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(4 + \sqrt{3}\)[/tex] is [tex]\(4 - \sqrt{3}\)[/tex]. Here are the steps:

1. Multiply by the conjugate:
[tex]\[ \frac{26}{4 + \sqrt{3}} \times \frac{4 - \sqrt{3}}{4 - \sqrt{3}} \][/tex]

2. Numerator multiplication:
- Expand the numerator:
[tex]\[ 26 \times (4 - \sqrt{3}) = 26 \times 4 - 26 \times \sqrt{3} = 104 - 26\sqrt{3} \][/tex]

3. Denominator multiplication:
- Use the difference of squares formula [tex]\( (a + b)(a - b) = a^2 - b^2\)[/tex]:
[tex]\[ (4 + \sqrt{3})(4 - \sqrt{3}) = 4^2 - (\sqrt{3})^2 = 16 - 3 = 13 \][/tex]

4. Rewrite the fraction:
[tex]\[ \frac{104 - 26\sqrt{3}}{13} \][/tex]

5. Simplify the fraction:
- Divide each term in the numerator by the denominator:
[tex]\[ \frac{104}{13} - \frac{26\sqrt{3}}{13} = 8 - 2\sqrt{3} \][/tex]

Thus, the rationalized form of [tex]\(\frac{26}{4 + \sqrt{3}}\)[/tex] is:
[tex]\[ 8 - 2\sqrt{3} \][/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.