Question
upstudy study bank question image url

Simplify, without using a calculator \( \left(\frac{\sqrt{7^{2011}}-\sqrt{7^{2009}}}{\sqrt{7^{2008}}}+\sqrt{7}\right)^{2} \)

Ask by Rojas Todd. in South Africa
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The simplified result is \(343\).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Mind Expander

To simplify the expression \(\left(\frac{\sqrt{7^{2011}}-\sqrt{7^{2009}}}{\sqrt{7^{2008}}}+\sqrt{7}\right)^{2}\), we start by rewriting the square roots: \[ \sqrt{7^{2011}} = 7^{1005.5}, \quad \sqrt{7^{2009}} = 7^{1004.5}, \quad \sqrt{7^{2008}} = 7^{1004} \] Now, we can express the original expression in terms of powers of 7. We rewrite the fraction: \[ \frac{\sqrt{7^{2011}} - \sqrt{7^{2009}}}{\sqrt{7^{2008}}} = \frac{7^{1005.5} - 7^{1004.5}}{7^{1004}} = \frac{7^{1004}(7 - 1)}{7^{1004}} = 7 - 1 = 6 \] Next, we can substitute this back into the original expression: \[ \frac{\sqrt{7^{2011}} - \sqrt{7^{2009}}}{\sqrt{7^{2008}}} + \sqrt{7} = 6 + \sqrt{7} \] Now we take this result and square it: \[ \left(6 + \sqrt{7}\right)^{2} \] Using the binomial expansion \((a + b)^2 = a^2 + 2ab + b^2\), we calculate: \[ = 6^2 + 2 \cdot 6 \cdot \sqrt{7} + (\sqrt{7})^2 = 36 + 12\sqrt{7} + 7 = 43 + 12\sqrt{7} \] Thus, the final simplified expression is: \[ \boxed{43 + 12\sqrt{7}} \]

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy