Higgins Tucker
03/18/2023 · Junior High School
Rationalise \( \frac{-5}{9-4 \sqrt{2}} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To rationalize the expression \( \frac{-5}{9 - 4\sqrt{2}} \), we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 9 - 4\sqrt{2} \) is \( 9 + 4\sqrt{2} \).
Let's perform the multiplication:
\[
\frac{-5}{9 - 4\sqrt{2}} \cdot \frac{9 + 4\sqrt{2}}{9 + 4\sqrt{2}} = \frac{-5(9 + 4\sqrt{2})}{(9 - 4\sqrt{2})(9 + 4\sqrt{2})}
\]
Now, we calculate the denominator using the difference of squares formula:
\[
(9 - 4\sqrt{2})(9 + 4\sqrt{2}) = 9^2 - (4\sqrt{2})^2 = 81 - 32 = 49
\]
Next, we calculate the numerator:
\[
-5(9 + 4\sqrt{2}) = -45 - 20\sqrt{2}
\]
Putting it all together, we have:
\[
\frac{-5(9 + 4\sqrt{2})}{(9 - 4\sqrt{2})(9 + 4\sqrt{2})} = \frac{-45 - 20\sqrt{2}}{49}
\]
Thus, the rationalized form of the expression is:
\[
\frac{-45 - 20\sqrt{2}}{49}
\]
This can also be expressed as:
\[
-\frac{45}{49} - \frac{20\sqrt{2}}{49}
\]
So, the final answer is:
\[
\frac{-45 - 20\sqrt{2}}{49}
\]
Quick Answer
\[
\frac{-45 - 20\sqrt{2}}{49}
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit