Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Simplify. Rationalize the denominator. \newline62+3\frac{6}{2 + \sqrt{3}}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Simplify radical expressions using conjugatesright chevron icon

Full solution

Q. Simplify. Rationalize the denominator. \newline62+3\frac{6}{2 + \sqrt{3}}
  1. Identify conjugate of denominator: Identify the conjugate of the denominator 2+32 + \sqrt{3}. The conjugate of a+ba + \sqrt{b} is a−ba - \sqrt{b}. Therefore, the conjugate of 2+32 + \sqrt{3} is 2−32 - \sqrt{3}.
  2. Multiply by conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the fraction by 11 in the form of the conjugate over itself: 2−32−3\frac{2 - \sqrt{3}}{2 - \sqrt{3}}.\newlineThis gives us 6×(2−3)(2+3)×(2−3)\frac{6 \times (2 - \sqrt{3})}{(2 + \sqrt{3}) \times (2 - \sqrt{3})}.
  3. Simplify numerator by distributing: Simplify the numerator by distributing the 66. \newline6×(2−3)6 \times (2 - \sqrt{3}) equals 12−6×312 - 6 \times \sqrt{3}.
  4. Simplify denominator using formula: Simplify the denominator by using the difference of squares formula.\newline(2+3)×(2−3)(2 + \sqrt{3}) \times (2 - \sqrt{3}) equals 22−(3)22^2 - (\sqrt{3})^2.\newlineThis simplifies to 4−34 - 3, which equals 11.
  5. Write simplified fraction: Write the simplified fraction.\newlineSince the denominator is now 11, the fraction simplifies to just the numerator: 12−6×312 - 6 \times \sqrt{3}.

More problems from Simplify radical expressions using conjugates

Question
Find the real-number root. \newline16\sqrt{16}\newline\newlineWrite your answer in simplified form. \newline____\_\_\_\_
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the real-number root. \newline916\sqrt{\frac{9}{16}}\newlineSimplify your answer and write it as a decimal or as a proper or improper fraction.\newline_____\_\_\_\_\_
Get tutor helpright-arrow

Posted 2 months ago

Question
Find the real-number root using a calculator. Round your answer to the nearest thousandth. \newline`\sqrt{957} `\newline _______
Get tutor helpright-arrow

Posted 2 months ago

Question
Multiply. Write your answer in simplest form. \newline273×42\sqrt{273} \times \sqrt{42}\newline______
Get tutor helpright-arrow

Posted 1 month ago

Question
Simplify. \newline634\sqrt{\frac{63}{4}}\newline______
Get tutor helpright-arrow

Posted 2 months ago

Question
Simplify. Rationalize the denominator.\newline−49−2\frac{-4}{9 - \sqrt{2}}\newline______
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve for z z .\newline16=z 16 = \sqrt{z} \newlinez= z = _____
Get tutor helpright-arrow

Posted 2 months ago

Question
Simplify the radical expression. 14x14 \sqrt{14x^{14}} Write your answer in the form A A , B \sqrt{B} , or AB A\sqrt{B} , where A A and B B are constants or expressions in x x . Use at most one radical in your answer, and at most one absolute value symbol in your expression for A A . ______
Get tutor helpright-arrow

Posted 2 months ago

Question
Simplify the radical expression.\newline12x12\sqrt{12x^{12}}\newlineWrite your answer in the form AA, B\sqrt{B}, or ABA\sqrt{B}, where AA and BB are constants or expressions in xx. Use at most one radical in your answer, and at most one absolute value symbol in your expression for AA.\newline‾\underline{\hspace{3cm}}
Get tutor helpright-arrow

Posted 2 months ago

Question
What is the value of sin⁡(5π6)? \sin \left(\frac{5 \pi}{6}\right) ? \newlineChoose 11 answer:\newline(A) −32 -\frac{\sqrt{3}}{2} \newline(B) −22 -\frac{\sqrt{2}}{2} \newline(C) 12 \frac{1}{2} \newline(D) 150150
Get tutor helpright-arrow

Posted 4 months ago

View all (60)