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. \newline93+2\frac{9}{3 + \sqrt{2}}

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. \newline93+2\frac{9}{3 + \sqrt{2}}
  1. Identify Conjugate of Denominator: Identify the conjugate of the denominator.\newlineThe conjugate of a number of the form a+ba + \sqrt{b} is a−ba - \sqrt{b}. Therefore, the conjugate of 3+23 + \sqrt{2} is 3−23 - \sqrt{2}.
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator.\newline(9×(3−2))/((3+2)×(3−2))(9 \times (3 - \sqrt{2})) / ((3 + \sqrt{2}) \times (3 - \sqrt{2}))
  3. Simplify Numerator: Simplify the numerator.\newlineMultiply 99 by each term in the conjugate.\newline9×3−9×29 \times 3 - 9 \times \sqrt{2}\newline= 27−9×227 - 9 \times \sqrt{2}
  4. Simplify Denominator: Simplify the denominator.\newlineUse the difference of squares formula, which states that (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.\newline(3+2)∗(3−2)=32−(2)2(3 + \sqrt{2}) * (3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2\newline=9−2= 9 - 2\newline=7= 7
  5. Write Final Expression: Write the simplified expression.\newlineNow that we have simplified both the numerator and the denominator, we can write the final expression.\newline(27−9×2)/7(27 - 9 \times \sqrt{2}) / 7\newlineThis fraction is already in simplest form.

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)